Marie Thévenon

Professeur Agrégé

A T The

THE CHRONICLES OF AN AUTHOR AND HIS CHARACTERS' AGEING FORETOLD

The Invention of Solitude in 1982 up until that of Winter Journal in 2012, Paul Auster, the prolific contemporary New York writer, has evoked, in his autobiographical and autofictional work, the life stories of several of his characters who grow old simultaneously with the author himself. It is in two of his most recent novels that the theme of old age is specifically addressed. Travels in the Scriptorium (2006) and Man in the Dark (2008) both evoke the physical and mental decline that two ageing characters suffer from. Both are writers approaching the end of their lives. These two novels form a diptych: the first happens in one day, the second, in one night.

The main character in Travels in the Scriptorium, Mr. Blank, is condemned by his condition as an "old man" to inaction and the impossibility of finding any form of coherent thought. Handicapped by old age and illness, he has serious problems of physical mobility. Moreover, he suffers from a form of senile dementia which leads him to forget his own identity. As he suffers enormously from this situation, he seeks refuge by withdrawing within himself, into a "locked room", a recurrent theme in Paul Auster's works.

The hero of Man in the Dark, August Brill, is also a prisoner within a "locked room" following an accident which has deprived him of the usage of his legs. Widowed, grieving and wounded by life, he must find refuge within his imagination in order to resist falling into the abyss of memories from his bygone life.

These two characters do not however intend to suffer passively the effects of their disabilities and decide to work diligently on their mind and imagination in the hope of improving their overall state. It is this paradox represented by their situation that will be studied in this article, a paradox

born from the surprising contrast between, on the one hand, the physical and mental incapacity caused by old age and, on the other hand, the positive effects of work undertaken on the mind and imagination which allows these disabled characters to obtain a surprising and quite unexpected mental virtuosity.

1) The representation of progressive ageing which leads to illness and death a) Characters who age at the same time as their author

In Paul Auster's works, the narrating characters are submitted, just like the author himself, to the natural process of ageing from one novel to the next. His first prose work, The Invention of Solitude, sets the tone by introducing an autobiographical summary of the life of the author, then aged 35. In mourning for his recently deceased father, the author chooses to describe exactly how he feels. This first prose work, devoted to the sudden unexpected death of his father, will have an influence on all his following work. The narrators of the novels which come after The Invention of Solitude often share the same date of birth as the author: 1947. Marco Stanley Fogg, the narrator of Moon Palace, explains on the very first page of the novel that he was 18 in 1965. As François Gavillon notes in Paul Auster, Gravité et légèreté de l'écriture 1 , a few temporal clues allow us to determine that 1947 is also Aaron's date of birth, making him the same age as Fogg and their common author 2 . Peter Aaron, whose initials recall Paul Auster's, is the narrator of Leviathan. David Zimmer in The Book of Illusions states his age as fifty-one at the moment when he writes his final manuscript, in January 1999 3 , that is to say approximately the same age as the author when he published his novel in 2002. In Oracle Night, Sidney Orr is introduced as a thirty-four year old man that a serious illness has transformed into an old man ("I was only thirty-four, but for all intents and purposes the illness had turned me into an old man", 1). However, since the story is told twenty years after this event (the footnote on page 1 states: "Twenty years have elapsed since that morning", 8), it seems that Sidney is fifty-four at the time of narration, which corresponds to Auster's age at the time his novel was published in 2003. As for John Trause, yet another writer-character in Oracle Night, and whose name "Trause" is an anagram of Auster's name, he is fifty-six when he dies ("[M]y friend was dead at fifty-six", 206), which is precisely the age of Paul Auster when he was writing Oracle Night. The Brooklyn Follies, published in 2005, features a narrator whose age is yet again close to Paul Auster's: "I hadn't been back [to Brooklyn] in fifty-six years [...]. My parents had moved out of the city when I was three" (1). Nathan Glass is thus fifty-nine, that is to say "almost sixty years old" (3), as he confirms two pages later.

Nathalie Cochoy observes that this game is less autobiographical than revealing of the way in which writing is oriented towards life4 . Auster's characters age at the same time as their author who chooses again and again to represent characters whose age is more or less the same as his. Now that this had been established, let us observe how this ageing process is not without its consequences.

b) Staging of old age and illness

From the third page of Travels in the Scriptorium, the narrator states that the adjective "old" applies to any person whose age is between sixty and a hundred years old 5 . This new and difficult stage in life is often accompanied by illness which imposes extra hardship with its harmful effects. Indeed, old age and illness are often associated in Paul Auster's works. Another distressing observation is that his ageing and ailing characters realise that their illness is progressing at the same time as their age. In some instances, it is their very illness which, by imposing itself forcefully, transforms a healthy young man into a prematurely aged person.

The leitmotiv of illness appears in The Book of Illusions (2002) in the shape of the sudden brutal onset of disease with no forewarning: a heart attack. This theme recurs often in Paul Auster's novels, as a reminder of his father's death because of a heart attack before he had even entered old age: "One day there is life. A man, for example, in the best of health, not even old, with no history of illness. [...] And then, suddenly, it happens there is death. A man lets out a little sigh, he slumps down in his chair, and it is death. [...] Death without warning. Which is to say: life stops. And it can stop at any moment" (The Invention of Solitude, 5). David Zimmer in The Book of Illusions, Joyce's husband, Stanley's wife and Nathan's symptoms6 in The Brooklyn Follies, Ed Victory's second wife and Ed Victory himself in Oracle Night are all examples of characters who succumb to heart attacks.

Other illnesses affect the narrator characters in Paul Auster's works: Sidney Orr transforms into an old man prematurely when he is thirty-four in Oracle Night. He no longer knows who he is and can hardly walk. In the same novel, John Trause, aged fifty-six, dies of a pulmonary embolism. In The Brooklyn Follies, Nathan Glass is in remission from cancer after having nearly died.

Travels in the Scriptorium and Man in the Dark stand out from the previous novels however in that they introduce characters who, contrary to those that we have mentioned up until now, are older than sixty years old and suffer from a handicap linked to actual old age.

In Travels in the Scriptorium, the situation which is introduced in the very first paragraphs presents an "old man" who has forgotten who he is and does not know where he is. As he tries to recollect this information, he is faced with the traumatic experience of trying to reactivate his memory, but discovering that his efforts are all in vain and that he is incapable of answering the most elementary questions. We are given very little initial information concerning this character. His physical description is very vague: "the old man's age, for example, is difficult to determine from the slightly out-of-focus black-and-white images. The only fact that can be set down with any certainty is that he is not young" (2-3). His very name is unknown and it is decided that he will be called by a name which symbolises his absence of identity: "Mr. Blank". He is Mr. Nothing, nothingness, the blank page of the author who lacks inspiration. The French translation of the novel indicates in a footnote that "Blank signifies white, like a white page, or empty, like a blank look, an empty box on a questionnaire"7 . In the absence of memory, the only clues available to Mr. Blank are visual and emotional. He does not know where he is or for how long he has been there, but he is suffering from a deep feeling of guilt. His solipsism is thus absolute at the beginning of this novel and old age and illness unite in order to erase his identity and disrupt his physical and mental capacities.

Man in the Dark presents a similar situation: the narrator never leaves the enclosed space inside his home which he considers as a rampart against the exterior world. Leaving this safe space would be extremely difficult for him anyway since pain in one of his legs (which he refers to as "my shattered leg", 44) because of a car accident forces him to stay confined to bed in his bedroom:

I would love to be able to walk up the stairs, to go to her room, and talk to her for a while. […] But I can't climb the stairs in a wheelchair, can I? And if I used my crutch, I would probably fall in the dark. Damn this idiot leg. The only solution is to sprout a pair of wings, giant wings of the softest white down. Then I'd be up there in a flash. (12)(13) At the age of 72, the narrator would like to be master of his own body but he knows that it is now but an unrealisable dream. Moreover, he suffers from other aches and pains, in particular a cough-which never ceases to punctuate his narrative-caused by his life as a smoker. On a psychological level, he is subject to the regular resurgence of memories from his past life as well as the obsessive memory of the recent death of his wife. His memory, which is particularly active during his nights of insomnia, however fails to function properly when it is needed for the most simple everyday events: "Odd that those figures should have stuck, but with so many other things slipping away from me now" (81).

As these narrating characters grow older, their preoccupations turn naturally towards problems linked to old age and death, even more so since, as we have seen, old age is linked to illness in Paul Auster's works. Starting with The Book of Illusions (2002), and in particular in Travels in the Scriptorium and Man in the Dark, the characters suffer from serious illnesses which deprive them of their mobility, rendering them dependent and forcing them to live as recluses. It is in this sinister context that the Austerian "locked room" is put in place. A place of reclusion, solitude, despair, but also a place of convalescence and rebirth, the "locked room" plays an ambiguous role in Travels in the Scriptorium and in Man in the Dark.

2) The Austerian locked room: a place of despair and of rebirth a) Duality of solitude within the locked room Solitude occupies an important place in Paul Auster's works as attests the very title of his first prose work, The Invention of Solitude. This theme of solitude leads to various and often opposed interpretations, in particular in French where the term can be used in two different ways, as opposed to the English language which offers the choice between "solitude" and "loneliness" 8 . However, the connotations of these two words vary a lot depending on the individual author. Paul Auster chooses the word "solitude" following the example of an author he often evokes, Henry David Thoreau, who presents the choice of solitude as a vital need: "I never found the companion that was so companionable as solitude. We are for the most part more lonely when we go abroad among men than when we stay in our chambers" (Walden,135).

Ever since The Invention of Solitude, the notion of solitude is analysed thoroughly in the light of Paul Auster's reading and personal experience. In his first novel, we encounter two forms of solitude which will often reappear in Auster's works: the negative experience of a harmful form of solitude, that of the father who lives isolated from the world, without experiencing any benefit from his isolation, and, on the other hand, the positive experience of an auto-imposed solitude, that of the son, which allows the character to open himself up to literary creation. Throughout The Invention of Solitude, Auster never ceases to oppose the prison cell type room where consciousness explodes and the place of creation it can become once such a space has been "colonised". "All the unhappiness of man stems from one thing only: that he is incapable of staying quietly in his room" 9 : as Pascal suggests, it is the individual himself who chooses to 8 The theologian and philosopher Paul Tillich explains well both aspects of solitude, whose most common definition tends to not mention its positive aspects: "Our language has wisely sensed these two sides of man's being alone. It has created the word 'loneliness' to express the pain of being alone. And it has created the word 'solitude' to express the glory of being alone. Although, in daily life, we do not always distinguish these words, we should do so consistently and thus deepen our understanding of our human predicament" (The Eternal Now, chapter 1, II).

Winter Chronicles: The Chronicles of an Author and his Characters' Ageing Foretold abandon himself to a sterile and harmful solitude, or to use this solitude as a step towards creation. "The Locked Room", which is the title of one of the three stories within The New York Trilogy, is the space which certain characters choose to isolate themselves from the exterior world in order to exclude themselves from society so as to focus on themselves. By locking himself in this enclosed space with the aim of finding a lost intimacy between mind and body, the hero must abandon all his spatiotemporal landmarks, leave the "here and now" and lose himself in the enclosed space, explains Dominique Coquet 10 . Mind and body merge in order to form but one big space of memory: " Memory as a room, as a body, as a skull that encloses the room in which a body sits" (The Invention of Solitude, 93). The "locked room" allows the character to delve within himself for better or for worse.

The locked room thus appears like a space of quiet solitude, like that of some of Vermeer's paintings, evoked by Paul Auster in The Invention of Solitude, which represent a woman in a room, alone with her solitude and her everyday gestures. Travels in the Scriptorium presents from the very beginning, in the manner of the Dutch painter, an old man sitting on the edge of his bed, with his head down staring at the floor, the image of a solitude which is impossible to decode in the first few lines of the novel. But the locked room can also be a paradoxical space in which the calm of a desired solitude and the horror inspired by imprisonment alternate, not unlike Van Gogh's Bedroom in Arles (1888): calm and relaxing at first, the painting awakens, according to Paul Auster, a sensation of imprisonment in a room without an exit. Finally, the locked room can also be a place of madness, like Hölderlin's tower. Fortunately, the comment made by Paul Auster in his first prose work about this third aspect of the "locked room" symbolised by the tower is not as negative as it seems: Auster doesn't focus so much on the sinister and desperate aspect of such a situation as that of Hölderlin's but considers, on the contrary, that the poet's imprisonment is, in a way, what saved him: "More than likely, it is the room that restored Hölderlin to life, that gave him back whatever life it 9 "[T]out le malheur des hommes vient d'une seule chose, qui est de ne savoir pas demeurer en repos dans une chambre". Quoted in English by Paul Auster in The Invention of Solitude, page 80. 10 Translated from French: " le héros doit abandonner ses repères spatio-temporels, décrocher du "ici et maintenant", et se perdre dans l'espace". Dominique Coquet, Identité de l'écriture et écriture de l'identité dans the New York Trilogy de Paul Auster. Thèse de doctorat sous la direction de Sylviane Burner, soutenue le 9 décembre 2005 à l'Université Paul Verlaine, Metz. Page 58.

was left for him to live" (106). It is a similar situation for Jonas, swallowed by the whale, who finds refuge within its stomach. Following this interpretation, it is possible to consider that Mr. Blank, by being subjected to this stay in the "locked room" will find his salvation rather than his perdition.

We will now study the way in which Mr. Blank and August Brill who both first have to endure the effects of a harmful solitude, the outcome of which can be nothing but death, manage to turn their ordeal into a positive experience which initiates a true rebirth and leads the characters onto the path of literary creation.

b) A place of imprisonment and despair

A large part of Travels in the Scriptorium is dedicated to an interactive relationship between Mr. Blank and the locked room. The importance of the setting of this novel is indicated from the third paragraph onwards. The space in which Mr. Blank finds himself resembles in all aspects the other Austerian "locked rooms" which we find frequently in his works and it possesses its usual characteristics: it is small and bare. However, as opposed to the other Austerian characters who have had to spend a period of time confined in such a room, Mr. Blank wonders throughout the novel why he is imprisoned there: is he detained there against his will or is he free to leave his room whenever he wishes to do so? This uncertainty plays an important role since Mr. Blank interprets his misadventure differently depending on whether he believes himself to be imprisoned or whether he thinks he is free to leave whenever he wants to. The ambiguity of this situation is maintained permanently because he will never manage to know the truth. In fact, he has no reason to leave his room since his physical state forces him to rest and to be dependent on others: he is no longer in control of his movements, he is prone to shaking, can hardly stand up, eat or wash himself without the help of a nurse. The ambiguity may remain regarding his status within the locked room, but what is clear is that he is imprisoned within his own defective body and lost mind. His situation is even more serious in that to his sense of being lost is added an impression of not being able to escape from his fate because of his great age. Such a character, with his old age and his helplessness and confusion, could be interpreted as a phantasmagorical projection of the ageing author, trapped by his failing body and wandering mind with no hope of being able to escape.

The setting, as we have seen, plays an important role and the first paragraphs of the novel immerse us immediately in a Beckettian atmosphere of emptiness: the old man is in the middle of a nearly empty room which contains merely a bed, a desk, a chair and a lamp. This room reminds us of the one in which "Dr. B." is imprisoned in Royal Game (Schachnovelle) by Stefan Zweig.

The arrangement of the furniture in the room as well as the relationship between the character and each object are of utmost importance throughout Travels in the Scriptorium. It is not a matter of chance if the first object mentioned is the bed on which Mr. Blank is sitting. A symbol of rest but also of illness and convalescence, such a piece of furniture plays a crucial role in the endless loop at the end of the novel which shows Mr. Blank lying down in his bed whilst the lights are switched off and a voice can be heard saying-and this will be the last sentence of the novel-: "Sleep well, Mr. Blank. Lights out" (130). Hence, everything starts on this bed and everything finishes there: we have come full circle, the end of the novel comes back to the very first scene in a vicious circle which is just as oppressing as the atmosphere of the room from which Mr. Blank will never escape.

The other piece of furniture, the desk, a symbol of creativity in most Austerian novels, takes here a different form. Indeed, in Auster's works, narration goes through the act of writing, a true ritual which is dutifully staged in each novel, showing the writer at his desk. It is at this Austerian desk that narration, which is usually in the first person, is written. Yet the desk cannot play the same role in Travels in the Scriptorium as in Auster's previous novels since this story is told in the third person. Mr. Blank is not the narrating character and it is thus not he who is telling and writing the story that the reader is reading. Hence, we must ask ourselves what the signification of this desk, which, for the first time in an Austerian work, is not used for the act of writing, is. In fact, it seems that this item of furniture is but an empty shell, the reminder of a past life as a writer, just as Mr. Blank is no longer more than the empty shell of a man who, from what the other characters have led him to believe, was once a prolific writer. The state of illness in which Mr. Blank finds himself does not allow him to use this desk to write and this symbolic piece of furniture seems to only be present as a representation of the character's past which seems, at first sight, definitely bygone.

The other novel which is the object of our study, Man in the Dark, introduces a character, August Brill, who is in what seems to be a desperate situation: he is an old man who is trapped in bed because of an injury to one of his legs. Mourning and guilt-ridden with remorse because of his past mistakes, he can no longer sleep at night. From the second page of the novel, he explains that he is trying to forget his sad situation by "telling himself stories" at night, when insomnia torments him. It is this form of escape, which he practises by appealing to his imagination, which will allow this character, who is bed-ridden because of his illness, to live, in parallel to his reclusive life, the adventures of a character emanating from his imagination, who takes part in a civil war and finds himself at the heart of the action. The enclosed space and the exterior world are thus opposed but join together through the first narration which generates the second in a vertical hierarchy allowing us to see the relationship between the two spaces, one interior and the other exterior.

Just like August Brill in Man in the Dark, Mr. Blank will manage to escape from the space in which he is trapped thanks to the work on his imagination.

c) A place of rebirth through the work of the mind

The few objects which are present in the "locked room" will play an important role in the recovery, by the character, of his memory, his identity and his physical and mental capacities, all the more so since there is not one instance when he is not using one of them.

As we have seen, one of the items of furniture which is contained in the room is the famous desk of the writer which reappears in each Austerian novel. If it is here rendered unusable as the place of writing because of Mr. Blank's physical state, it still plays an important role since upon it are placed some objects which function as clues which will help Mr. Blank to reactivate his memory and remember certain people he has met in his past life. Mr. Blank thus finds on this desk thirty-six photos which will allow him to associate faces with names which are given to him, reminding us of the research done by Paul Auster in The Invention of Solitude in order to find members of his own family.

Other objects are also on the desk: a notebook in which Mr. Blank starts taking notes throughout the day, writing down the names of characters as and when he hears them. This quite rudimentary note-taking constitutes a first step towards the reappropriation of memory. Next to the notebook are four piles of paper. The one Mr. Blank gives the most importance to in the course of the story is a manuscript which seems to have been written by a prisoner of war waiting for the moment of his execution. After having read a few paragraphs, Mr. Blank cannot determine whether it is a work of fiction or an authentic report, a true testimony. A quick stylistic analysis convinces him that it is a text that was written in the 19 th century and which offers the peculiarity of being unfinished, which greatly annoys Mr. Blank. One of the characters who is following Mr. Blank's progress, a certain Dr. Fanshawe, asks him to invent the rest and we thus witness this work of creation until the end of the novel.

And that is how, thanks to this exercise in writing, Mr. Blank rediscovers little by little his memory as well as his identity as a writer. Story-writing allows him to control time and thus to restructure life. Indeed, being able to once again distinguish between past, present and future allows him to give meaning to life. Writing-or at least the process of structuring a story, since in this instance Mr. Blank does not write but works in the manner of a writer-as we can see, plays an essential role in this process of reconstruction by allowing the character to regain the power of awakening his memory and thus to reconquer his mind and body which recover their capacities.

In Travels in the Scriptorium, the objects are connecting-documents which become identity clues allowing Mr. Blank to reappropriate his lost memory, in the same way that in The Invention of Solitude photos and abandoned objects belonging to Auster's father helped the writer to reconstruct his identity as a lone man and to fit into his family by rediscovering a story which had many links with that of other members of that family and which gave it all meaning. As Auster says in The Invention of Solitude: "[T]he moment we step into the space of memory, we walk into the world" (178). Like a child who invents a world in order to play, the old man rebuilds his world with the help of his imagination which enables him to find a new youth.

The stay in the locked room can thus have a double effect. The first is to condemn the person who is in there to a definite mortiferous and sterile imprisonment. The second is to allow the rebirth of the individual since well-managed isolation can take on a positive aspect, trigger a rediscovery of one's self and render the individual available for creative activity. As Pascal Bruckner observes:

For Auster, confinement is a form of exile. [...] This penchant for narrow spaces, where the spirit can project itself against the walls [...] makes the room a kind of mental uterus, site of a second birth. In this enclosure the subject gives birth, in essence, to himself. From mere biological existence he now attains spiritual life. [...] The self must die, Auster seems to say, in order to live [...]. (28) It is in such an instance that the locked room as a physical space merges with the locked room as the mental space of the character. It truly becomes a uterus, a place of rebirth. This is an image which reappears in The Brooklyn Follies when Tom Wood's very small flat is described: "the womblike dark of his tiny home" (131). In this space resolutely situated "outside the world", death and rebirth merge into one as Wood's room is also compared to a coffin ("the dismal coffin he called his room", 132). Indeed, rebirth can only happen after the death of the self: "The self must die [...] in order to live". In order for this double operation of death and rebirth to take place, the locked room must belong to another dimension and form a private, timeless space, a space of absolute solipsism in which the Austerian character, or Auster himself, finds himself face to face with himself. Others can no longer see him. He disappears, in a sense, from the world.

The locked room does not really belong to life itself but to its very beginning and its end ("womblike", "coffin"). The rebirth of the characters who seek refuge within this interior space happens symbolically in the other novels during nine months and it is of utmost significance that this rebirth takes the same amount of time as the gestation period leading to birth. The comparison to the womb thus takes on its full meaning: it is at the end of the gestation period that the Austerian narrator "gives birth" to the work which he began nine months before, and leaves his place of confinement, ready to rediscover life from a different angle. As a place of foetal "regression", the locked room allows the Austerian characters to experience a true rebirth.

The end of Travels in the Scriptorium and Man in the Dark is not as optimistic as that of Auster's other novels since we witness an uninterrupted succession of days of reclusion in the former and that the latter presents an elderly insomniac who continues to suffer just as much from his insomnia at the end of the novel. Yet, even if these characters see no improvement in their painful situation, they seem less desperate at the end of the novel. The work of creation they have undertaken throughout their narration has helped them to leave behind, for a little while, their state as an old man and has allowed them to regain the intellectual vivacity which they thought they had lost. This mental work allows them to find once again a virtuosity which is capable of overcoming their handicap and to gain, at the very least, some compensation for the physical loss which old age imposes on us.

François Gavillon, Paul Auster, Gravité et légèreté de l'écriture. Rennes : Presses universitaires de Rennes,

2 Translated from French: "Quelques indices temporels permettent de fixer à 1947 l'année de naissance de Aaron, qui a donc le même âge que Fogg et que leur auteur commun" (172)3 "I turned fifty-one in March 1998. [...] They sent me home from the hospital in mid-December, and by early January I was writing the first pages of this book" (The Book ofIllusions,[317][318]

Translated from French: "[c]e jeu est moins autobiographique que révélateur de la manière dont l'écriture s'oriente vers la vie". Nathalie Cochoy, Passante à New York. Bordeaux : Presses Universitaires de Bordeaux, 2009. Page 194.

"the word old is a flexible term and can be used to describe a person anywhere between sixty and a hundred" (Travels in the Scriptorium, 3)

This traumatic experience is directly inspired by a real misadventure of Paul Auster's who was the victim of an inflamed oesophagus, the symptoms of which are very similar to those of a heart attack, and hence thought that he was about to die, just like Nathan Glass in The Brooklyn Follies (Cf. Winter Journal : "your false heart attack", 31-32).

Translated from the French: "Blank signifie blanc, comme une page blanche, ou vide, comme un regard vide, une case vide dans un questionnaire"(11)


Jean-Claude Bermond

Xavier Muñoz

Alberto Marchetti-Spaccamela

Induced broadcasting algorithms in iterated line digraphs

come   L'archive ouverte pluridisciplinaire


Jean-Franc ¸ois Coulombel
email: jean-francois.coulombel@math.univ-lille1.fr

Antoine Gloria
email: antoine.gloria@inria.fr

SEMIGROUP STABILITY OF FINITE DIFFERENCE SCHEMES FOR MULTIDIMENSIONAL HYPERBOLIC INITIAL BOUNDARY VALUE PROBLEMS

Keywords: 1991 Mathematics Subject Classification. Primary: 65M12; Secondary: 65M06, 35L50. Key words and phrases. Hyperbolic systems, boundary value problems, finite difference schemes, stability

We develop a simple energy method to prove the stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems. In particular we extend to several space dimensions a crucial result by Goldberg and Tadmor. This allows us to give two conditions on the discretized operator that ensure that stability estimates for zero initial data imply a semigroup stability estimate for general initial data. We then apply this criterion to several numerical schemes in two space dimensions.

Introduction

The aim of this article is to prove stability estimates for finite difference discretizations of hyperbolic initial boundary value problems. A general theory to derive such estimates has been developed in [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF] for one-dimensional problems and later extended in [START_REF] Michelson | Stability theory of difference approximations for multidimensional initial-boundary value problems[END_REF] to multidimensional problems. The analysis for the discretized equations is similar to the theory in [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF] for the continuous problem (namely for hyperbolic systems of partial differential equations), and relies on the so-called normal modes analysis. Due to the fact that the method uses a Laplace transform in time, the estimates in [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF][START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF][START_REF] Michelson | Stability theory of difference approximations for multidimensional initial-boundary value problems[END_REF] are restricted to zero initial data. A natural question is then to show that problems that are stable for zero initial data are also stable for non-zero initial data. Making the space of "suitable" initial data precise is part of the question.

For continuous problems, this question was solved in [START_REF] Rauch | L 2 is a continuable initial condition for Kreiss' mixed problems[END_REF]. (We also refer to [1, chapter 4] for a complete description of the results.) For discretized problems, the question was solved in [13] where the author proves stability for non-zero initial data in one space dimension. The proof in [13] relies on a crucial Lemma (Lemma 2.3 in [START_REF] Goldberg | Scheme-independent stability criteria for difference approximations of hyperbolic initialboundary value problems[END_REF]), that we shall refer to as Goldberg-Tadmor's Lemma. Goldberg-Tadmor's Lemma shows that the Dirichlet boundary conditions yield stable problems for discretized scalar equations and zero initial data. Consequently, it is not clear whether Goldberg-Tadmor's Lemma, and therefore the results of [13], extends to hyperbolic systems in several space dimensions (because systems usually do not decouple into a collection of scalar equations). In this article, we develop a simple energy method with which we recover Goldberg-Tadmor's Lemma and the stability results of [13] and that is flexible enough to handle discretized multidimensional systems. Opposite to the original proof in [START_REF] Goldberg | Scheme-independent stability criteria for difference approximations of hyperbolic initialboundary value problems[END_REF], our new proof of Goldberg-Tadmor's Lemma covers the case of non-zero initial data in ℓ 2 . Once we have obtained a stability estimate for Dirichlet boundary conditions and non-zero initial data, it is almost straightforward to show that discretizations that are stable for zero initial data are also stable for non-zero initial data. Note that, although this is an old problem, Theorem 3.1 below seems to be the first general stability result for discretized initial boundary problems in several space dimensions.

We first develop our approach on one-dimensional problems. The one-dimensional analysis is extended to multidimensional problems in Section 3 using a partial Fourier transform in the tangential space variables. Then in Section 4, we give examples of discretizations to which our analysis applies.

Notation. In all this paper, we let M d,D (K) denote the set of d × D matrices with entries in K = R or C, and we use the notation M D (K) when d = D. If M ∈ M D (C), sp(M ) denotes the spectrum of M , ρ(M ) denotes the spectral radius of M , while M * denotes the conjugate transpose of M . We let I denote the identity matrix, without mentioning the dimension. The norm of a vector x ∈ C D is |x| := (x * x) 1/2 . Eventually, we let ℓ 2 denote the set of square integrable sequences, without mentioning the indices of the sequences (sequences may be valued in C d for some integer d).

The letter C denotes a constant that may vary from line to line or within the same line. The dependence of the constants on the various parameters is made precise throughout the text.

One-dimensional problems

For one-dimensional problems, we introduce the following notation for norms on ℓ 2 (Z). Let ∆x > 0 be a space step. For all integers m 1 ≤ m 2 , we set

u 2 m1,m2 := ∆x m2 j=m1 |u j | 2
to denote the ℓ 2 -norm on the interval [m 1 , m 2 ] (m 1 may equal -∞ and m 2 may equal +∞). The corresponding scalar product is denoted by (•, •) m1,m2 .

2.1. Main result in one space dimension. We consider a hyperbolic initial boundary value problem in one space dimension:

     ∂ t u + A ∂ x u = F (t, x) , (t, x) ∈ R + × R + , B u(t, 0) = g(t) , t ∈ R + , u(0, x) = f (x) , x ∈ R + , (1)
where A ∈ M D (R) is diagonalizable with real eigenvalues, and B ∈ M D+,D (R) with D + the number of positive eigenvalues of A (counted with their multiplicity). We assume that the boundary is noncharacteristic, that is 0 ∈ sp(A). Problem (1) is well-posed in any suitable sense if and only if:

R D = Ker B ⊕ E + (A) ,
where E + (A) is the unstable eigenspace of A (associated with positive eigenvalues of A). In that case, the solution u to (1) belongs to C (R + ; L 2 (R + )) and its trace on {x = 0} is well-defined and belongs to e γ t L 2 (R + ) for all γ > 0. Moreover, for every parameter γ > 0, u satisfies the energy estimate:

sup t≥0 e -2 γ t u(t, •) 2 L 2 (R + ) + γ +∞ 0 e -2 γ t u(t, •) 2 L 2 (R + ) dt + +∞ 0 e -2 γ t |u(t, 0)| 2 dt ≤ C f 2 L 2 (R + ) + 1 γ +∞ 0 e -2 γ t F (t, •) 2 L 2 (R + ) dt + +∞ 0 e -2 γ t |g(t)| 2 dt , (2)
where the constant C is independent of γ, f, F, g. The estimate (2) can be localized on any finite time interval [0, T ] because the solutions to (1) satisfy a causality principle ("future does not affect the past"). We now introduce the finite difference approximation of (1). Let ∆x, ∆t > 0 denote the space and time steps, where the ratio λ = ∆t/∆x is a fixed positive constant, and let p, q, r be some integers. The solution u to (1) is approximated by a sequence (U n j ) defined for n ∈ N, and j ∈ 1 -r + N. For j = 1 -r, . . . , 0, U n j approximates the trace u(n ∆t, 0) on the boundary {x = 0}, and possibly the trace of normal derivatives. The boundary meshes [j ∆x, (j + 1) ∆x[, j = 1 -r, . . . , 0, shrink to {0} as ∆x tends to 0. Hence the "formal" limit problem as ∆x tends to 0 is set on the half-line R + . We consider one-step finite difference approximations of (1) that read 1

:      U n+1 j = Q U n j + ∆t F n j , j ≥ 1 , n ≥ 0 , U n+1 j = B j,-1 U n+1 1 + B j,0 U n 1 + g n+1 j , j = 1 -r, . . . , 0 , n ≥ 0 , U 0 j = f j , j ≥ 1 -r , (3)
where the operators Q, B j,-1 , B j,0 are given by:

Q := p ℓ=-r A ℓ T ℓ , B j,σ := q ℓ=0 B ℓ,j,σ T ℓ , T ℓ U m k := U m k+ℓ . (4)
In ( 4), all matrices A ℓ , B ℓ,j,σ belong to M D (R) and depend on λ, A, B but not on ∆t (or equivalently ∆x). We recall the following definition from [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF]:

Definition 2.1 (Strong stability [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF]). The finite difference approximation (3) is said to be strongly stable if there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution (U n j ) of (3) with f = 0 satisfies the estimate:

γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t U n 2 1-r,+∞ + n≥0 ∆t e -2 γ n ∆t 0 j=1-r |U n j | 2 ≤ C    γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    .
The estimate in Definition 2.1 is the discrete counterpart of the energy estimate (2) for the "continuous" problem (1) in the case of zero initial data (and when one does not require to control the L ∞ t (L 2 x ) norm of the solution). We recall that strong stability in the sense of Definition 2.1 is usually proved by performing a Laplace transform with respect to the time variable. The energy estimate for the resolvent equation is then obtained by using symmetrizers whose construction relies on the so-called uniform Kreiss-Lopatinskii condition (non-existence of unstable nor weakly unstable normal modes). We refer to [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF] for some results in this direction. In this paper, we shall consider that the scheme (3) is strongly stable, and we wish to prove an energy estimate for (3) in the case of non-zero initial data. In view of (2), the most obvious space of initial data for (3) is ℓ 2 . Let us now introduce our main assumptions, and then state our result.

For ℓ = -r, . . . , p, and z ∈ C \ {0}, let us define the matrices:

A ℓ (z) := δ ℓ0 I - 1 z A ℓ , (5)
where δ ℓ1ℓ2 is the Kronecker symbol. We make the following assumption 2 :

Assumption 2.1. The matrix A p (z) is invertible for all z ∈ C with |z| ≥ 1.

Our second crucial assumption is the following:

Assumption 2.2. The operator Q satisfies Q v -∞,+∞ ≤ v -∞,+∞ for all v ∈ ℓ 2 .
Our main result is stated as follows:

Theorem 2.1. Let Assumptions 2.1 and 2.2 be satisfied, and assume that the scheme (3) is strongly stable in the sense of Definition 2.1. Then there exists a constant C such that for all γ > 0 and all 1 We do not focus here on the construction of such approximations and refer to [START_REF] Gustafsson | Time dependent problems and difference methods[END_REF] for some examples that enter this framework, see also Section 4.

2 Assumption 2.1 is similar to Assumption 5.5 in [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF].

∆t ∈ ]0, 1], the solution U to (3) satisfies the estimate:

sup n≥0 e -2 γ n ∆t U n 2 1-r,+∞ + γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t U n 2 1-r,+∞ + n≥0 ∆t e -2 γ n ∆t 0 j=1-r |U n j | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    . (6)
The method of proof is inspired from [13] with however some important modifications. More precisely, we shall introduce an auxiliary discretized problem where we modify the boundary operators B j,-1 , B j,0 . The auxiliary problem is chosen in such a way that even for non-zero initial data, the solution can be estimated by applying the energy method (in some sense, the auxiliary boundary conditions are "strictly dissipative"). Our auxiliary problem is not the same as in [13]. As a matter of fact, we shall show directly that the Dirichlet boundary conditions are strictly dissipative. As announced in the introduction, this is an improved version of Goldberg-Tadmor's Lemma. Our new proof of Goldberg-Tadmor's Lemma relies on the energy method and can therefore be extended to multidimensional problems even if the equation is not scalar (D ≥ 2), see Section 3. Once we have the estimate for the auxiliary boundary conditions, the end of the proof follows [13], see also the arguments in [1, chapter 4] for the continuous problem.

2.2.

A refined version of Goldberg-Tadmor's Lemma. In this paragraph, we consider the following auxiliary discretization where the (non-homogeneous) Dirichlet conditions are enforced at the boundary:

     V n+1 j = Q V n j + ∆t F n j , j ≥ 1 , n ≥ 0 , V n+1 j = g n+1 j , j = 1 -r, . . . , 0 , n ≥ 0 , V 0 j = f j , j ≥ 1 -r . (7)
The aim of this paragraph is to prove the following:

Theorem 2.2. Let Assumptions 2.1 and 2.2 be satisfied. Then there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution V to (7) satisfies the estimate:

sup n≥0 e -2 γ n ∆t V n 2 1-r,+∞ + γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t V n 2 1-r,+∞ + n≥0 ∆t e -2 γ n ∆t max(p,q+1) j=1-r |V n j | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    . (8)
In particular, the discretization (7) is strongly stable in the sense of Definition 2.1.

The key point in the proof of Theorem 2.2 is Assumption 2.2, which can be understood as a symmetry assumption on the matrices A ℓ . We emphasize that Assumption 2.2 can be extended to multidimensional systems, as we shall see in section 3, while the "scalar" assumption in [START_REF] Goldberg | Scheme-independent stability criteria for difference approximations of hyperbolic initialboundary value problems[END_REF] does not extend to general multidimensional systems.

The proof of Theorem 2.2 is split in several steps. We first observe that the solution V to (7) depends linearly on the source terms (f, g, F ). It is therefore sufficient to prove [START_REF] Rauch | L 2 is a continuable initial condition for Kreiss' mixed problems[END_REF] in the case F = 0 (no source term in the interior equation) and in the case (f, g) = 0 (zero initial data, and homogeneous boundary conditions). It turns out that we use slightly different arguments for the two cases. We begin with the case F = 0, and then treat the case (f, g) = 0.

We point out that in [START_REF] Rauch | L 2 is a continuable initial condition for Kreiss' mixed problems[END_REF], we estimate the (weighted) ℓ 2 -norm in time of the trace (V n j ) n≥0 , for all j from 1 -r to max(p, q + 1). As a matter of fact, it would have been sufficient for the proof of Theorem 2.1 to have this type of estimate up to j = q + 1. However, in the proof below, we shall first obtain an estimate up to j = p (see Corollary 2.1 below). This is the reason why we have stated [START_REF] Rauch | L 2 is a continuable initial condition for Kreiss' mixed problems[END_REF] in this way. We start the proof of Theorem 2.2 proper with a series of preliminary results: Lemma 2.1. Let Assumptions 2.1 and 2.2 be satisfied. Then there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution V to [START_REF] Michelson | Stability theory of difference approximations for multidimensional initial-boundary value problems[END_REF] with F = 0 satisfies the estimate:

e 2 γ ∆t sup n≥1 e -2 γ n ∆t V n 2 1,+∞ + γ n≥1 ∆t e -2 γ n ∆t V n 2 1,+∞ + n≥0 ∆t e -2 γ n ∆t p j=1-r |V n j | 2 ≤ C    f 2 1-r,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    . (9)
Proof of Lemma 2.1. We decompose the operator Q as:

Q := I + Q .
Then Assumption 2.2 is equivalent to the inequality:

∀ w ∈ ℓ 2 , 2 w, Q w -∞,+∞ + Q w 2 -∞,+∞ ≤ 0 . ( 10
)
We first use the relation V n+1 j = (I + Q) V n j for j ≥ 1 (recall that F = 0), and derive:

V n+1 2 1,+∞ -V n 2 1,+∞ = 2 V n , Q V n 1,+∞ + Q V n 2 1,+∞ . (11)
For a fixed integer n, we introduce the sequence (W j ) such that W j = V n j for j ≥ 1 -r and W j = 0 for j ≤ -r. Due to the structure of the operator Q, see (4), we have Q W j = 0 if j ≤ -r -p, and

Q W j = Q V n j if j ≥ 1.
Using [START_REF] Turkel | Symmetric hyperbolic difference schemes and matrix problems[END_REF], we thus get:

0 ≥ 2 W, Q W -∞,+∞ + Q W 2 -∞,+∞ = 2 V n , Q W 1-r,0 + 2 V n , Q V n 1,+∞ + Q W 2 1-r-p,-r + Q W 2 1-r,0 + Q V n 2 1,+∞ = 2 V n , Q V n 1,+∞ + Q V n 2 1,+∞ + V n + Q W 2 1-r,0 + Q W 2 1-r-p,-r -V n 2 1-r,0 . (12)
We insert ( 12) into ( 11) and obtain:

V n+1 2 1,+∞ -V n 2 1,+∞ + Q W 2 1-r-p,-r + V n + Q W 2 1-r,0 ≤ V n 2 1-r,0 . (13)
At this point, two situations may occur depending on p. Let us first consider the case p ≥ 1. Then, by Assumption 2.1, A p is an invertible matrix. We have the following: Lemma 2.2. Let p ≥ 1 and let A p be invertible. Then there exists a constant c > 0 that does not depend on ∆t nor on V n such that the following estimate holds:

Q W 2 1-r-p,-r + V n + Q W 2 1-r,0 ≥ c V n 2 1-r,p .
Let us assume that Lemma 2.2 holds and go back to (13). We have:

V n+1 2 1,+∞ -V n 2 1,+∞ + c ∆x p j=1-r |V n j | 2 ≤ ∆x 0 j=1-r |V n j | 2 . ( 14
)
The end of the proof consists in integrating (14) over N, see a similar calculation in the continous case in [1, page 95]. Let γ > 0, and for the sake of clarity, let us introduce the following notation:

V n := e -2 γ n ∆t V n 2 1,+∞ , B n := e -2 γ n ∆t p j=1-r |V n j | 2 , G n := e -2 γ n ∆t 0 j=1-r |V n j | 2 .
We multiply (14) by exp(-2 γ n ∆t):

e 2 γ ∆t V n+1 -V n + c λ ∆t B n ≤ 1 λ ∆t G n .
Summing this inequality from 0 to N yields:

e 2 γ ∆t V N +1 + e 2 γ ∆t -1 ∆t N 1 ∆t V n + c λ N 0 ∆t B n ≤ V 0 + 1 λ N 0 ∆t G n ≤ V 0 + 1 λ n≥0 ∆t G n .
Letting N tend to +∞, we have proved:

e 2 γ ∆t sup n≥1 V n + γ n≥1 ∆t V n + n≥0 ∆t B n ≤ C   V 0 + ∆x G 0 + n≥1 ∆t G n   , (15)
and the right-hand side of ( 15) is directly estimated by the right-hand side of ( 9), see the definition above for G n and use [START_REF] Michelson | Stability theory of difference approximations for multidimensional initial-boundary value problems[END_REF]. The constant C in (15) is independent of γ and ∆t and we have therefore completed the proof of ( 9) in the case p ≥ 1.

It remains to treat the case p = 0 for which Lemma 2.2 does not hold anymore. In this case, we go back to (13) and simply ignore the nonnegative "boundary terms" on the left-hand side:

V n+1 2 1,+∞ -V n 2 1,+∞ ≤ V n 2 1-r,0 .
Then we proceed as above (with the same notation) and derive the weighted-in-time estimate:

e 2 γ ∆t sup n≥1 V n + γ n≥1 ∆t V n ≤ C   V 0 + ∆x G 0 + n≥1 ∆t G n   .
In the case p = 0, the term:

n≥0 ∆t e -2 γ n ∆t p j=1-r |V n j | 2 ,
in the left-hand side of ( 9) is directly estimated by the right-hand side of (9) so the proof of Lemma 2.1 is complete (provided that we prove Lemma 2.2, which is done below).

Proof of Lemma 2.2. Proving Lemma 2.2 is equivalent to proving that the following quadratic form (that is independent on n):

(V n 1-r , . . . , V n p ) -→ -r j=1-r-p | Q W j | 2 + 0 j=1-r |V n j + Q W j | 2 , (16)
is positive definite. (Recall that W denotes the extension of V n by zero for j ≤ -r.) The quadratic form ( 16) is clearly nonnegative. Let us therefore consider some vector (V n 1-r , . . . , V n p ) that satisfies:

∀ j = 1 -r -p, . . . , -r, Q W j = 0 , ∀ j = 1 -r, . . . , 0, V n j + Q W j = 0 . ( 17
)
We first show by induction on j that V n j = 0 for all j = 1 -r, . . . , p -r. Let us recall that p ≥ 1, so we can write Q = Q -I under the form:

Q = A p T p + p-1 ℓ=-r A ℓ T ℓ .
In particular, we have

Q W 1-r-p = A p V n 1-r , so V n 1-r = 0 because A p is invertible. For j = 1 -r - p, . . . , -r, Q W j equals A p V n
j+p plus a linear combination of the V n ℓ , ℓ < j + p. Since the first term V n 1-r is zero, we can proceed by induction and we thus get

V n 1-r = • • • = V n p-r = 0.
We now use the second set of equalities in (17). In particular, we have

V n 1-r + Q W 1-r = Q W 1-r = A p V n 1-r+p .
We therefore get V n 1-r+p = 0, and the rest of the proof follows from another induction argument. We have therefore shown that (17) implies (V n 1-r , . . . , V n p ) = 0, so the quadratic form ( 16) is positive definite. The proof of Lemma 2.2 is complete.

We next turn to the case (f, g) = 0 and the interior source F is arbitrary. We have: Lemma 2.3. Let Assumptions 2.1 and 2.2 be satisfied. Then there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution (V n j ) to (7) with (f, g) = 0 satisfies the estimate:

sup n≥1 e -2 γ n ∆t V n 2 1,+∞ + γ γ ∆t + 1 n≥1 ∆t e -2 γ n ∆t V n 2 1,+∞ + e -2 γ ∆t n≥1 ∆t e -2 γ n ∆t p j=1 |V n j | 2 ≤ C γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ . ( 18
)
Proof of Lemma 2.3. Following the proof of Lemma 2.1, we decompose the operator Q as

Q = I + Q.
Then we use the relation V n+1 j = Q V n j + ∆t F n j for j ≥ 1 and derive:

V n+1 2 1,+∞ -V n 2 1,+∞ = 2 V n , Q V n 1,+∞ + Q V n 2 1,+∞ + 2 Q V n , F n 1,+∞ + ∆t 2 F n 2 1,+∞ .
Let us first assume that p is positive, so that Lemma 2.2 holds. Proceeding as in the proof of Lemma 2.1, we obtain the inequality (recall that here we have homogeneous boundary conditions):

V n+1 2 1,+∞ -V n 2 1,+∞ + c ∆x p j=1 |V n j | 2 ≤ 2 ∆t V n 1,+∞ F n 1,+∞ + ∆t 2 F n 2 1,+∞ . (19)
For the sake of clarity, we now introduce the notation:

V n := e -2 γ n ∆t V n 2 1,+∞ , B n := e -2 γ n ∆t p j=1 |V n j | 2 , F n := e -2 γ (n+1) ∆t F n 2 1,+∞ .
We multiply (19) by exp(-2 γ (n + 1) ∆t) and get:

V n+1 -e -2 γ ∆t V n + c λ e -2 γ ∆t ∆t B n ≤ 2 ∆t e -γ ∆t F 1/2 n V 1/2 n + ∆t 2 F n .
Summing this inequality from 0 to N and recalling that the initial data is zero, we obtain:

V N +1 + 1 -e -2 γ ∆t ∆t N 1 ∆t V n + c λ e -2 γ ∆t N 0 ∆t B n ≤ ∆t N 0 ∆t F n + C N 0 ∆t F n ∆t V n .
We apply Young's inequality for the last term on the right-hand side and we end up with:

V N +1 + 1 -e -2 γ ∆t 2 ∆t N 0 ∆t V n + c λ e -2 γ ∆t N 0 ∆t B n ≤ C ∆t 1 -e -2 γ ∆t N 0 ∆t F n .
Letting N tend to +∞, we have proved (recall that the initial data is zero):

sup n≥1 V n + γ γ ∆t + 1 n≥1 ∆t V n + e -2 γ ∆t n≥1 ∆t B n ≤ C γ ∆t + 1 γ n≥0 ∆t F n .
The constant C is independent of γ and ∆t.

The case p = 0 is dealt with in an entirely similar way. In this case, the term:

n≥1 ∆t e -2 γ n ∆t p j=1 |V n j | 2
that appears on the left-hand side of (18) vanishes. The proof of Lemma 2.3 is thus complete.

If we compare Lemma 2.1 to Lemma 2.3, we observe that in (18) the estimate for the trace (V n j ) n≥0 , j = 1, . . . , p, involves a factor exp(-2 γ ∆t) that deteriorates the estimate when γ ∆t is large. We are going to derive an additional estimate that will enable us to get rid of this factor. This is done in: Lemma 2.4. Let Assumptions 2.1 and 2.2 be satisfied. Then there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1] verifying γ ∆t ≥ 1, the solution (V n j ) to (7) with (f, g) = 0 satisfies the estimate:

n≥1 ∆t e -2 γ n ∆t p j=1 |V n j | 2 ≤ C γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ . (20)
Proof of Lemma 2.4. Let j ∈ {1, . . . , p}. We use the equation V n+1 j = Q V n j + ∆t F n j , and derive:

|V n+1 j | 2 ≤ 2 |Q V n j | 2 + 2 ∆t 2 |F n j | 2 ≤ C 1 ∆t V n 2 1,+∞ + ∆t F n 2 1,+∞
.

We multiply this inequality by exp(-2 γ (n + 1) ∆t) and sum with respect to n ≥ 0:

n≥1 ∆t e -2 γ n ∆t |V n j | 2 ≤ C   e -2 γ ∆t ∆t n≥1 ∆t e -2 γ n ∆t V n 2 1,+∞ + ∆t n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞   .
We now use Lemma 2.3 to estimate the first term on the right-hand side of the inequality. We obtain:

n≥1 ∆t e -2 γ n ∆t |V n j | 2 ≤ C e -2 γ ∆t γ ∆t + 1 γ ∆t + 1 γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ ,
and the result follows.

The proof of Lemma 2.4 is surprisingly simple, but unfortunately it does cover the small values of γ ∆t. If we collect Lemma 2.1, Lemma 2.3 and Lemma 2.4, we obtain: Corollary 2.1. Let Assumptions 2.1 and 2.2 be satisfied. Then there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution V to (7) satisfies the estimate:

sup n≥0 e -2 γ n ∆t V n 2 1-r,+∞ + γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t V n 2 1-r,+∞ + n≥0 ∆t e -2 γ n ∆t p j=1-r |V n j | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    . ( 21
)
Proof of Corollary 2.1. First of all, Lemma 2.1 shows that (21) holds when the interior source term F vanishes. Indeed, (9) implies the weaker inequality:

sup n≥1 e -2 γ n ∆t V n 2 1,+∞ + γ γ ∆t + 1 n≥1 ∆t e -2 γ n ∆t V n 2 1,+∞ + n≥0 ∆t e -2 γ n ∆t p j=1-r |V n j | 2 ≤ C    f 2 1-r,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    . (22)
To obtain (21) in the case F = 0, it remains to argue that one may replace sup n≥1 e -2 γ n ∆t V n 2 1,+∞ in (22) by sup n≥1 e -2 γ n ∆t V n 2 1-r,+∞ , and add the term

γ γ∆t + 1 n≥1 ∆te -2γn∆t V n 2 1-r,0 = 1 λ γ∆t γ∆t + 1 n≥1 ∆t e -2γn∆t 0 j=1-r |V n j | 2
to the left handside. To this aim, it is enough to note that

V n 2 1-r,0 = 0 j=1-r ∆x |V n j | 2 = λ ∆t 0 j=1-r |g n j | 2 ,
which precisely appears in the right handside of (22). Changing C accordingly proves the claim.

When the initial data and the boundary source term vanish, the estimate (21) is obtained by combining Lemma 2.3 (when γ ∆t ∈ ]0, 1]) and Lemma 2.4 (when γ ∆t ≥ 1). We need not recover the boundary terms in the supremum as above since these boundary terms vanish here.

If we compare the result of Corollary 2.1 with [13, Theorem 3.2], we get a better information when p is greater than 2, since we get additional trace estimates. Another advantage of our approach is that in [13], the author uses Goldberg-Tadmor's Lemma after introducing his auxiliary boundary operator. Hence, in some sense he needs two auxiliary problems. Here we shall only deal with the original discretization (3) and the auxiliary discretization [START_REF] Michelson | Stability theory of difference approximations for multidimensional initial-boundary value problems[END_REF].

Corollary 2.1 gives the result of Theorem 2.2 when p > q. For q ≥ p, we need some additional trace estimates. These estimates can be obtained by adapting the method described in [13, page 85]. As a matter of fact, we use simpler arguments than in [13], which are only based on the energy method. In particular, we nowhere refer to the results of [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF].

End of the proof of Theorem 2.2. From now on, we consider the case q ≥ p since for q < p, Corollary 2.1 gives the result of Theorem 2.2. Once again, the proof of ( 9) is slightly different according to the value of p. Let us first assume p ≥ 1. As in [13], we define the sequence W n j := V n j+1 for n ≥ 0 and j ≥ 1 -r. Then (W n j ) solves the system:

         W n+1 j = Q W n j + ∆t F n j+1 , j ≥ 1 , n ≥ 0 , W n+1 j = g n+1 j+1 , j = 1 -r, . . . , -1 , n ≥ 0 , W n+1 0 = V n+1 1 , n ≥ 0 , W 0 j = f j+1 , j ≥ 1 -r .
We can apply Corollary 2.1 to W and obtain:

n≥0 ∆t e -2 γ n ∆t |W n p | 2 ≤ C    f 2 2-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 2,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=2-r |g n j | 2 + n≥1 ∆t e -2 γ n ∆t |V n 1 | 2    .
We use Corollary 2.1 again to estimate the last term on the right-hand (this is possible because p ≥ 1), and we get:

n≥0 ∆t e -2 γ n ∆t |V n p+1 | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    .
We have therefore derived a trace estimate for (V n p+1 ) n≥0 . A straightforward induction argument gives:

n≥0 ∆t e -2 γ n ∆t q+1 j=p+1 |V n j | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    . (23)
Combining ( 21) and ( 23), we obtain (9) for the case p ≥ 1.

We now consider the case p = 0, for which Corollary 2.1 does not give any trace estimate of (V n j ) n≥0 with j ≥ 1. Hence, we can not use the shift argument of [13] as above. Using Assumption 2.1, we know that the spectral radius of A 0 is strictly less than 1. Consequently, there exists a positive definite symmetric matrix H and there exists a positive number ε 0 such that if we consider the new Euclidean norm on R D :

∀ X ∈ R D , |X| H := X; H X , then we have: ∀ X ∈ R D , |A 0 X| H ≤ √ 1 -2 ε 0 |X| H .
We start from the relation:

V n+1 1 = A 0 V n 1 + -1 ℓ=-r A ℓ V n 1+ℓ + ∆t F n 1 = A 0 V n 1 + 0 j=1-r A j-1 g n j + ∆t F n 1 =:X n ,
where we use the notation g 0 j := f j for j = 1 -r, . . . , 0. Then we derive:

|V n+1 1 | 2 H = |A 0 V n 1 | 2 H + 2 A 0 V n 1 ; H X n + |X n | 2 H ≤ (1 -2 ε 0 ) |V n 1 | 2 H + 2 A 0 V n 1 ; H X n + |X n | 2 H ≤ (1 -ε 0 ) |V n 1 | 2 H + (1 + ε -1 0 ) |X n | 2 H .
Using the definition of X n , the latter inequality gives:

|V n+1 1 | 2 H -|V n 1 | 2 H + ε 0 |V n 1 | 2 H ≤ C   ∆t F n 2 1,+∞ + 0 j=1-r |g n j | 2   .
Using the same summation process as earlier, we obtain:

(1 -e -2 γ ∆t ) + ε 0 e -2 γ ∆t n≥0 ∆t e -2 γ n ∆t |V n 1 | 2 H ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    .
The norm | • | H and the standard Euclidean norm are equivalent, so we get:

n≥0 ∆t e -2 γ n ∆t |V n 1 | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    , (24)
with a constant C that does not depend on γ nor on ∆t. The proof of (9) follows from an induction argument where we apply the above method to recover the estimate for the trace (V n j ) n≥0 , j = 2, . . . , q + 1. The proof of Theorem 2.2 is now complete.

2.3.

Proof of Theorem 2.1. We decompose the solution U to (3) as U = V +W , where V satisfies:

     V n+1 j = Q V n j + ∆t F n j , j ≥ 1 , n ≥ 0 , V n+1 j = g n+1 j , j = 1 -r, . . . , 0 , n ≥ 0 , V 0 j = f j , j ≥ 1 -r , (25)
and W satisfies:

     W n+1 j = Q W n j , j ≥ 1 , n ≥ 0 , W n+1 j = B j,-1 W n+1 1 + B j,0 W n 1 + g n+1 j , j = 1 -r, . . . , 0 , n ≥ 0 , W 0 j = 0 , j ≥ 1 -r . (26)
The source term g in ( 26) is defined by:

∀ j = 1 -r, . . . , 0 , ∀ n ≥ 1 , g n j := B j,-1 V n 1 + B j,0 V n-1 1 . ( 27
)
The estimate of V is given by Theorem 2.2. Moreover, the discretization ( 3) is strongly stable in the sense of Definition 2.1, so W satisfies:

γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t W n 2 1-r,+∞ + n≥0 ∆t e -2 γ n ∆t 0 j=1-r |W n j | 2 ≤ C n≥1 ∆t e -2 γ n ∆t 0 j=1-r | g n j | 2 .
Here, we use the fact that the initial data for (26) is zero. The estimate of g n j is straightforward using the definition ( 27) and ( 9). In the end, we obtain:

γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t W n 2 1-r,+∞ + n≥0 ∆t e -2 γ n ∆t 0 j=1-r |W n j | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    . (28)
If we combine the estimate (28) for W and the estimate (9) for V , we see that the final task in the proof of Theorem 2.1 is to control the ℓ ∞ n (ℓ 2 j ) norm of W . We recall the following result that is the analogue of [13, Lemma 3.1]: Lemma 2.5. Let Assumptions 2.1 and 2.2 be satisfied, and assume that the discretization (3) is strongly stable. Then there exists a constant C that does not depend on the data g such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution W to (26) satisfies:

n≥0 ∆t e -2 γ n ∆t q+1 j=1 |W n j | 2 ≤ C n≥1 ∆t e -2 γ n ∆t 0 j=1-r | g n j | 2 .
We give a proof of Lemma 2.5 in Appendix A. As a matter of fact, Lemma 2.5 was stated without a proof in [13], although the result does not appear exactly in this form in [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF]. We find it useful to give a complete and detailed proof here. Using Lemma 2.5, let us rewrite (26) as:

     W n+1 j = Q W n j , j ≥ 1 , n ≥ 0 , W n+1 j = G n+1 j , j = 1 -r, . . . , 0 , n ≥ 0 , W 0 j = 0 , j ≥ 1 -r , (29)
with an obvious definition for the source term G. We apply Theorem 2.2 to (29) and obtain:

sup n≥0 e -2 γ n ∆t W n 2 1-r,+∞ ≤ C n≥1 ∆t e -2 γ n ∆t 0 j=1-r |G n j | 2 .
We now combine Lemma 2.5 and our previous estimate of g (see the argument above to get (28)), to derive:

n≥1 ∆t e -2 γ n ∆t 0 j=1-r |G n j | 2 ≤ C    f 2 1-r,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j=1-r |g n j | 2    .
We thus get the expected estimate for the ℓ ∞ n (ℓ 2 j ) norm of W , and the proof of Theorem 2.1 is complete.

Multidimensional problems

For multidimensional problems, we need further notation for norms on ℓ 2 (Z d ). Let ∆x i > 0 for i = 1, . . . , d be d space steps. For all integers m 1 ≤ m 2 , we set:

|||u||| 2 m1,m2 := ∆x 1 m2 j1=m1 d k=2 ∆x k d i=2 ji∈Z |u j1,...,j d | 2 , to denote the ℓ 2 -norm on the set [m 1 , m 2 ] × Z d-1 (m 1 may equal -∞ and m 2 may equal +∞).
We shall also make use of the ℓ 2 (Z d-1 )-norm that we denote by • : for all v ∈ ℓ 2 (Z d-1 ),

v 2 := d k=2 ∆x k d i=2 ji∈Z |v j2,...,j d | 2 .
3.1. Main result in several space dimensions. We consider the hyperbolic initial boundary value problem corresponding to (1) in several space dimensions d > 1, that is:

           ∂ t u + d i=1 A i ∂ xi u = F (t, x) , (t, x) ∈ R + × R d + , B u(t, (0, x ′ )) = g(t, x ′ ) , t ∈ R + , x ′ ∈ R d-1 , u(0, x) = f (x) , x ∈ R d + , (30)
where

R d + := R + × R d-1 , the matrices A i ∈ M D (R) are such that the symbol R d ∋ ξ → A(ξ) := d i=1 ξ i A i is uniformly diagonalizable (see [1, Theorem 1.3]
) in R, and B ∈ M D+,D (R) with D + the number of positive eigenvalues of A 1 . We assume that the boundary is noncharacteristic, that is 0 ∈ sp(A 1 ). Problem (30) is strongly well-posed in L 2 (R d + ) if and only if the matrices {A i } and B satisfy the so-called uniform Kreiss-Lopatinskii condition. In that case, the solution u to (30) belongs to C (R + ; L 2 (R d + )) and its trace on {x 1 = 0} is well-defined and belongs to e γ t L 2 (R + ; L 2 (R d-1 )) for all γ > 0. Moreover, for all γ > 0, u satisfies the energy estimate:

sup t≥0 e -2 γ t u(t, •) 2 L 2 (R d + ) + γ +∞ 0 e -2 γ t u(t, •) 2 L 2 (R d + ) dt + +∞ 0 e -2 γ t u(t, (0, •)) 2 L 2 (R d-1 ) dt ≤ C f 2 L 2 (R d + ) + 1 γ +∞ 0 e -2 γ t F (t, •) 2 L 2 (R d + ) dt + +∞ 0 e -2 γ t g(t, •) 2 L 2 (R d-1 ) dt , ( 31
)
where the constant C is independent of γ, f, F, g.

As for the one-dimensional case, we introduce the finite difference approximation of (30). We denote by ∆x := {∆x i } i=1,...,d and ∆t the space and time steps related by the fixed ratios λ i = ∆t/∆x i . For all j ∈ Z d , we set j = (j 1 , j ′ ) with j ′ = (j 2 , . . . , j d ). We let p, q, r ∈ N d be some multi-integers, and define p 1 , q 1 , r 1 , p ′ , q ′ , r ′ according to the above notation. The solution u to (30) is approximated by a sequence (U n j ) = (U n j1,j ′ ) for n ∈ N, j 1 ∈ 1 -r 1 + N, and j ′ ∈ Z d-1 . For j 1 = 1 -r 1 , . . . , 0, U n j1,• approximates the trace u(n∆t, 0, •) on the boundary {x 1 = 0}, and possibly the trace of normal derivatives. We consider one-step finite difference approximations of (30) that read:

     U n+1 j1,j ′ = Q U n j1,j ′ + ∆t F n j1,j ′ , j 1 ≥ 1 , j ′ ∈ Z d-1 , n ≥ 0 , U n+1 j1,j ′ = B j1,-1 U n+1 1,j ′ + B j1,0 U n 1,j ′ + g n+1 j1,j ′ , j 1 = 1 -r 1 , . . . , 0 , j ′ ∈ Z d-1 , n ≥ 0 , U 0 j1,j ′ = f j1,j ′ , j 1 ≥ 1 -r 1 , j ′ ∈ Z d-1 , (32)
where the operators Q, B j1,-1 , B j1,0 are given by:

Q := p1 ℓ1=-r1 p ′ ℓ ′ =-r ′ A ℓ1,ℓ ′ T ′ ℓ ′ T ℓ1 , B j1,σ := q1 ℓ1=0 q ′ ℓ ′ =-q ′ B ℓ1,ℓ ′ ,j1,σ T ′ ℓ ′ T ℓ1 , T ℓ1 U m k1,k ′ := U m k1+ℓ1,k ′ , T ′ ℓ ′ U m k1,k ′ := U m k1,k ′ +ℓ ′ . (33)
In ( 33), all matrices A ℓ , B ℓ,j1,σ belong to M D (R) and depend on {λ i , A i } i=1,...,d , B but not on ∆t (or equivalently not on ∆x). For multidimensional problems, the notion of strong stability now reads: Definition 3.1 (Strong stability [START_REF] Michelson | Stability theory of difference approximations for multidimensional initial-boundary value problems[END_REF]). The finite difference approximation (32) is said to be strongly stable if there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution (U n j ) of (3) with f = 0 satisfies the estimate:

γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t |||U n ||| 2 1-r1,+∞ + n≥0 ∆t e -2 γ n ∆t 0 j1=1-r1 U n j1,• 2 ≤ C    γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t |||F n ||| 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j1=1-r1 g n j1,• 2    .
We are in position to introduce the hypotheses corresponding to Assumptions 2.1 and 2.2 in the multidimensional case. For ℓ 1 = -r 1 , . . . , p 1 , and z ∈ C \ {0}, let us define the linear mappings:

A ℓ1 (z) : ℓ 2 (Z d-1 ) → ℓ 2 (Z d-1 ) w → δ ℓ10 w - 1 z p ′ ℓ ′ =-r ′ A ℓ1,ℓ ′ T ′ ℓ ′ w . (34)
We make the following first assumption:

Assumption 3.1. The mapping A p1 (z) is coercive on ℓ 2 (Z d-1
) for all z ∈ C with |z| ≥ 1. More precisely, there exists a constant c > 0 such that for all z ∈ C with |z| ≥ 1 and for all w ∈ ℓ 2 (Z d-1 ), we have:

A p1 (z)w ≥ c |z| ν w , ν := 1 if p 1 > 0, 0 otherwise. Remark 3.1.
If p 1 = 0, then Assumption 3.1 amounts to assuming that A 0 (z) is an isomorphism on ℓ 2 (Z d-1 ) for all |z| ≥ 1. For p 1 > 0, Assumption 3.1 is slightly weaker than assuming that A p1 (z) is an isomorphism since the fulfillment of Assumption 3.1 does not necessarily imply the surjectivity of A p1 (z) on ℓ 2 (Z d-1 ).

Our second assumption is unchanged:

Assumption 3.2. The operator Q satisfies |||Q v||| -∞,+∞ ≤ |||v||| -∞,+∞ for all v ∈ ℓ 2 (Z d ).
Our main stability result corresponding to Theorem 2.1 then reads: Theorem 3.1. Let Assumptions 3.1 and 3.2 be satisfied, and assume that the scheme (32) is strongly stable in the sense of Definition 3.1. Then there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution U to (32) satisfies the estimate:

sup n≥0 e -2 γ n ∆t |||U n ||| 2 1-r1,+∞ + γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t |||U n ||| 2 1-r1,+∞ + n≥0 ∆t e -2 γ n ∆t 0 j1=1-r1 U n j1,• 2 ≤ C    |||f ||| 2 1-r1,+∞ + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t |||F n ||| 2 1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j1=1-r1 g n j1,• 2    .
Our proof of Theorem 3.1 relies on the application of the one-dimensional analysis. In particular, we shall see below why Assumptions 3.1 and 3.2 are the natural extensions of Assumptions 2.1 and 2.2 in several space dimensions.

Proof of Theorem 3.1. Let us start from the scheme (32) that defines the sequence (U n j ). For each pair (j 1 , n) with j 1 ≥ 1 -r 1 and n ≥ 0, we define the piecewise constant function

V n j1 (x ′ ) on R d-1 such that V n j1 (x ′
) equals U n j1,j ′ on the mesh element with index j ′ . In a similar way, we define the functions F n j1 , g n j1 , f j1 . Applying Fourier transform with respect to the tangential variables x ′ in (32), we obtain:

       V n+1 j1 (ξ ′ ) = Q(θ ′ ) V n j1 (ξ ′ ) + ∆t F n j1 (ξ ′ ) , j 1 ≥ 1 , n ≥ 0 , V n+1 j1 (ξ ′ ) = B j1,-1 (θ ′ ) V n+1 1 (ξ ′ ) + B j1,0 (θ ′ ) V n 1 (ξ ′ ) + g n+1 j1 (ξ ′ ) , j 1 = 1 -r 1 , . . . , 0 , n ≥ 0 , V 0 j1 (ξ ′ ) = f j1 (ξ ′ ) , j 1 ≥ 1 -r 1 , (35)
where ξ ′ = (ξ 2 , . . . , ξ d ) denotes the frequency variables, θ ′ is a short notation for (ξ 2 ∆x 2 , . . . , ξ d ∆x d ), and the operators Q(θ ′ ), B j1,-1 (θ ′ ), B j1,0 (θ ′ ) are defined by:

Q(θ ′ ) := p1 ℓ1=-r1 p ′ ℓ ′ =-r ′ A ℓ1,ℓ ′ e i ℓ ′ •θ ′ T ℓ1 , B j1,σ (θ ′ ) := q1 ℓ1=0 q ′ ℓ ′ =-q ′ B ℓ1,ℓ ′ ,j1,σ e i ℓ ′ •θ ′ T ℓ1 .
We are reduced to a collection of one-dimensional problems parametrized by the frequencies ξ ′ . The end of the proof is based on the following observation: Lemma 3.1. Let Assumptions 3.1 and 3.2 be satisfied, and assume that the scheme (32) is strongly stable in the sense of Definition 3.1. Then for all ξ ′ ∈ R d-1 , the scheme (35) satisfies Assumptions 2.1 and 2.2, and is strongly stable in the sense of Definition 2.1. Moreover, the estimate of Theorem 2.1 holds with a constant C that is independent of ξ ′ .

The proof of Lemma 3.1 is performed below. With the help of Lemma 3.1, let us now complete the proof of Theorem 3.1. We can apply Theorem 2.1 to the scheme (35) with a constant C that is independent of ξ ′ . We thus obtain:

sup n≥0 e -2 γ n ∆t V n (ξ ′ ) 2 1-r1,+∞ + γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t V n (ξ ′ ) 2 1-r1,+∞ + n≥0 ∆t e -2 γ n ∆t 0 j1=1-r1 | V n j1 (ξ ′ )| 2 ≤ C    f (ξ ′ ) 2 1-r,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j1=1-r1 | g n j1 (ξ ′ )| 2 + γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t F n (ξ ′ ) 2 1,+∞    . (36)
Fubini's and Plancherel's Theorems yield

sup n≥0 e -2 γ n ∆t |||U n ||| 2 1-r1,+∞ = sup n≥0 e -2 γ n ∆t R d-1 V n (x ′ ) 2 1-r1,+∞ dx ′ = 1 (2 π) d-1 sup n≥0 e -2 γ n ∆t R d-1 V n (ξ ′ ) 2 1-r1,+∞ dξ ′ ≤ 1 (2 π) d-1 R d-1 sup n≥0 e -2 γ n ∆t V n (ξ ′ ) 2 1-r1,+∞ dξ ′ ,
after switching the order of the supremum and of the integral. We then integrate (36) with respect to ξ ′ ∈ R d-1 , use Fubini's and Plancherel's Theorem. Combined with the latter inequality, this proves Theorem 3.1. (Observe that it is crucial here that the right handside of (6) only contains ℓ 2 type norms so that we may apply Plancherel's Theorem, while the ℓ ∞ -norm only appears on the left handside of [START_REF] Lax | Difference schemes for hyperbolic equations with high order of accuracy[END_REF].) The proof of Theorem 3.1 is therefore complete provided we prove Lemma 3.1, which is done below.

Proof of Lemma 3.1. Let us first show that Assumption 2.2 is satisfied. We know that the norm of Q as an operator on ℓ 2 (Z d ) is not greater than 1. Using Plancherel's Theorem, this property is equivalent to the fact that the norm of the symbol Q of Q is not greater than 1. Decomposing frequencies θ ∈ R d as θ = (θ 1 , θ ′ ), the symbol Q is given by:

Q(θ) = p1 ℓ1=-r1 p ′ ℓ ′ =-r ′ A ℓ1,ℓ ′ e i ℓ ′ •θ ′ e i ℓ1 θ1 .
In other words, the symbol of the operator Q(θ ′ ) is nothing but Q(•, θ ′ ). This shows that the norm of Q(θ ′ ) as an operator on ℓ 2 (Z) is not greater than 1, and Assumption 2.2 is satisfied.

Let us turn to Assumption 2.1. With slight abuse of notation, if w ∈ ℓ 2 (Z d-1 ), we still denote w the piecewise constant function defined on R d-1 , and whose value on the mesh element with index j ′ equals w j ′ . Using Plancherel's Theorem, we have:

A p1 (z)w 2 = 1 (2 π) d-1 R d-1 |( A p1 (z)w)(ξ ′ )| 2 dξ ′ = 1 (2 π) d-1 R d-1 |A p1 (z, θ ′ ) w(ξ ′ )| 2 dξ ′ ,
where θ ′ is again a short notation for (ξ 2 ∆x 2 , . . . , ξ d ∆x d ), and where the matrices A p1 (z, θ ′ ) are defined by:

A p1 (z, θ ′ ) := δ p10 I - 1 z p ′ ℓ ′ =-r ′ A p1,ℓ ′ e i ℓ ′ •θ ′ , (37)
see (34). The operator A p1 is coercive if and only if the matrices A p1 (z, θ ′ ) are invertible for all z ∈ C with |z| ≥ 1 and all θ ′ ∈ R d-1 . This proves that Assumption 2.1 is satisfied for all θ ′ ∈ R d-1 . Next we argue that the constants in Theorem 2.1 are independent of ξ ′ . A close look at the proof shows that the only places where the constants may depend on ξ ′ are Lemma 2.2 and (24). On the one hand, Assumption 3.1 provides a lower bound for the coercivity constants of A p1 (z, θ ′ ) which is uniform in θ ′ , so that the constant in Lemma 2.2 is uniform in θ ′ . On the other hand, for p 1 = 0, we claim that the norm | • | H(θ ′ ) and the standard Euclidian norm | • | are equivalent uniformly in θ ′ so that the constant in (24) can be chosen independent of θ ′ . To this aim, it is enough to prove that sup

θ ′ ∈R d-1 |A 0 (1, θ ′ )| < ∞ , (38)
sup

θ ′ ∈R d-1 ρ(A 0 (1, θ ′ )) < 1 . (39)
Since θ ′ → A 0 (1, θ ′ ) is continuous and periodic on R d-1 , (38) is trivial, the supremum in (39) is attained and the bound follows from Assumption 3.1 and (37).

The only remaining task is to prove that the scheme ( 35) is strongly stable in the sense of Definition 2.1. This is done, as above, by applying Plancherel's Theorem to the strong stability estimate in Definition 3.1. We omit the details.

3.2.

A multidimensional version of Goldberg-Tadmor's lemma. The same argument as above (Fourier transform in the tangential variables and application of the one-dimensional results of Section 2) also applies to the case of nonhomogeneous Dirichlet boundary conditions. More precisely, let us consider the following auxiliary problem:

     V n+1 j1,j ′ = Q V n j1,j ′ + ∆t F n j1,j ′ , j 1 ≥ 1 , j ′ ∈ Z d-1 , n ≥ 0 , V n+1 j1,j ′ = g n+1 j1,j ′ , j 1 = 1 -r 1 , . . . , 0 , j ′ ∈ Z d-1 , n ≥ 0 , V 0 j1,j ′ = f j1,j ′ , j 1 ≥ 1 -r 1 , j ′ ∈ Z d-1 . (40)
The multidimensional version of Theorem 2.2 reads: Theorem 3.2. Let Assumptions 3.1 and 3.2 be satisfied. Then there exists a constant C such that for all γ > 0 and all ∆t ∈ ]0, 1], the solution V to (40) satisfies the estimate:

sup n≥0 e -2 γ n ∆t |||V n ||| 2 1-r1,+∞ + γ γ ∆t + 1 n≥0 ∆t e -2 γ n ∆t |||V n ||| 2 1-r1,+∞ + n≥0 ∆t e -2 γ n ∆t max(p1,q1+1) j1=1-r1 V n j1,• 2 ≤ C    |||f ||| 2 1-r1,+∞ + n≥1 ∆t e -2 γ n ∆t 0 j1=1-r1 g n j1,• 2
+ γ ∆t + 1 γ n≥0 ∆t e -2 γ (n+1) ∆t |||F n ||| 2 1,+∞    .
In particular, the discretization (40) is strongly stable in the sense of Definition 3.1.

We do not detail the proof of Theorem 3.2 since the arguments are entirely similar to the arguments used in the proof of Theorem 3.1 for passing from one-dimensional results to multidimensional results.

Examples and comments

4.1.

Examples and comments in one space dimension. For one-dimensional problems, the matrices A ℓ in the finite difference operator Q are usually polynomials of the matrix λ A where A is the matrix of the hyperbolic operator in (1) and λ = ∆t/∆x. If we assume furthermore that A is symmetric, then there exists an orthogonal matrix that diagonalizes simultaneously all the matrices A ℓ . In this case, Assumption 2.2 is exactly equivalent to the ℓ 2 -stability of the finite difference approximation, which is itself equivalent to the well-known von Neumann condition (see [3, chapter 5]):

∀ κ ∈ S 1 , ρ p ℓ=-r κ ℓ A ℓ ≤ 1 .
We recall that ρ denotes the spectral radius of a square matrix. Assumption 2.2 is therefore very reasonable and not restrictive in one space dimension. Assumption 2.1 is also satisfied in numerous situations and is rather easy to check.

We now make a few comments on the proof of Theorem 2.2. The proof in the case F = 0 follows from Lemma 2.1 and is done in a single way, whatever the values of γ and ∆t. However, we have seen that the case F = 0 requires two different approaches depending on the value of γ ∆t. Here, we report on a simple numerical test which shows that the two regimes γ ∆t ≤ 1 and γ ∆t ≥ 1 are different when F = 0. We consider the scalar transport equation:

∂ t u -∂ x u = F (t, x) , (t, x) ∈ R + × R + , u(0, x) = f (x) , x ∈ R + ,
where no boundary condition is required. We discretize the equation with the Lax-Friedrichs scheme and homogeneous Dirichlet boundary condition 3 :

     v n+1 j = (v n j-1 + v n j+1 )/2 + λ (v n j+1 -v n j-1 )/2 + ∆t F (n ∆t, j ∆x) , j ≥ 1 , n ≥ 0 , v n+1 0 = 0 , n ≥ 0 , v 0 j = f (j ∆x) , j ≥ 0 , (41)
or with the Lax-Wendroff scheme and homogeneous Dirichlet boundary condition:

     v n+1 j = v n j + λ (v n j+1 -v n j-1 )/2 + λ 2 (v n j+1 + v n j-1 -2 v n j )/2 + ∆t F (n ∆t, j ∆x) , j ≥ 1 , n ≥ 0 , v n+1 0 = 0 , n ≥ 0 , v 0 j = f (j ∆x) , j ≥ 0 . (42)
The CFL parameter λ is 0.9 in both situations, which ensures that Theorem 2.2 holds.

We plot the the ratio between the norm of the trace:

n≥0 ∆t e -2 γ n ∆t |v n 1 | 2 ,
and the right-hand side of (8) as a function of γ for the following cases:

(1) f (x) = 1 for x ∈ [0, 1], F = 0. The simulation is performed on the space interval [0, 2] with 1000 grid points. The parameter γ ranges from 10 -2 to 10 2 . (2) f = 0, F (t, x) = 1 for t ≥ 0 and x ∈ [0, 1]. The simulation is performed on the space interval [0, 2] with 1000 grid points. The parameter γ ranges from 10 -2 to 10 2 .

The results are plotted in Figure 1 for the Lax-Friedrichs scheme (41), and in Figure 2 for the Lax-Wendroff scheme (42). The observation is the following: the ratio between the norm of the trace and the norm of the source term depends monotonically on γ when f = 0 and F = 0, while it does not depend monotonically on γ when f = 0 and F = 0. This seems to indicate that in the case F = 0, the estimate (8) for small values of γ does not follow from the same arguments as for large values of γ. This is the reason why we believe that Lemmas 2.3 and 2.4 are both useful.

4.2.

Examples and comments in several space dimensions. Let us first comment on our multidimensional version of Golberg-Tadmor's lemma. In one dimension, the commutation assumption by Golberg and Tadmor is very natural, and essentially, it is "equivalent" to Assumption 2.2. The original work by Golberg and Tadmor [START_REF] Goldberg | Scheme-independent stability criteria for difference approximations of hyperbolic initialboundary value problems[END_REF] also covers the multidimensional case provided the matrices {A ℓ } in (33) commute. This assumption is very restrictive for d > 1 and amounts more or less to consider d uncoupled scalar equations in (30). Assumption 3.2, however, can hold independently of the fact that the matrices {A ℓ } do commute or not. Theorem 3.2 is therefore a true generalization of Golberg-Tadmor's lemma.

In the remaining part of this paragraph, we consider d = 2 and give several examples of discretizations to which Theorem 3.1 applies. 3 Observe that the homogeneous Dirichlet condition is not consistent in the L ∞ -norm with the continuous problem for which no boundary condition is required. However, we are concerned here with stability estimates, and consistency is a different issue. We consider the following two-dimensional problem:

     ∂ t u + A 1 ∂ x1 u + A 2 ∂ x2 u = F (t, x) , (t, x) ∈ R + × R 2 + , B u(t, 0, x ′ ) = g(t, x ′ ) , t ∈ R + , x ′ ∈ R, u(0, x) = f (x) , x ∈ R 2 + , (43)
where A 1 , A 2 and B are matrices, and A 1 and A 2 are symmetric. We assume that f ∈ L 2 (R 2 + ), and that there exists γ > 0 such that g ∈ e γt L 2 (R + ; L 2 (R)), and F ∈ e γt L 2 (R + ; L 2 (R 2 )). We let f j1,j2 , F n j1,j2 and g n j1,j2 be discrete approximations of f , F and g at time n∆t, and points (j 1 ∆x 1 , j 2 ∆x 2 ), where ∆t, ∆x 1 and ∆x 2 are the time and space steps related through the fixed ratios λ 1 = ∆x 1 /∆t and λ 2 = ∆x 2 /∆t. As before, U n j1,j2 denotes the approximation of the solution u to (43) at time n∆t, and points (j 1 ∆x 1 , j 2 ∆x 2 ). As for the one-dimensional examples (41) and (42), we address stability issues, not consistency, and we replace the boundary condition in (43) by Dirichlet boundary conditions for the numerical scheme. Let C 0 > 0 denote an arbitrary constant. Then for λ 2 ≤ C 0 λ 1 , we have:

A (ξ 2 ) = λ 1 A 1 + O(λ 2 1 ) ,
uniformly in ξ 2 ∈ R. Hence, Assumption 3.1 holds if det A 1 = 0 and λ 1 is small enough (the second CFL parameter λ 2 is subject to the restriction λ 2 ≤ C 0 λ 1 ). In this case, Theorem 3.2 shows the stability of the scheme (50) with Dirichlet boundary conditions and general initial data. Let us take a closer look at Assumption 3.1 on one specific example. To this aim, we consider:

A 1 = 1 0 0 -1 , A 2 = 0 1 1 0 .
These matrices satisfy A 2 1 = A 2 2 = I and A 1 A 2 + A 2 A 1 = 0, so that the symbol A above reduces to:

A (ξ 2 ) =    λ 1 + λ 2 1 -(λ 2 1 + λ 2 2 ) sin 2 ξ 2 2 0 0 -λ 1 + λ 2 1 -(λ 2 1 + λ 2 2 ) sin 2 ξ 2 2    .
Let X := sin 2 (ξ 2 /2) ∈ [0, 1]. The symbol A (ξ 2 ) is non-invertible if and only if λ 1 +λ 2 2 -(λ 2 1 +λ 2 2 ) X = 0, that is:

X = λ 1 + λ 2 1 λ 2 1 + λ 2 2 .
Since X ∈ [0, 1], this may only happen for λ 1 ≤ λ 2 2 . Hence, Assumption 3.1 is satisfied if and only if

λ 2 < λ 1 . (52)
Theorem 3.2 then shows that the scheme (50) with Dirichlet boundary conditions and general initial data is stable provided (51) and (52) hold. In this case, Assumption 3.1 does have an impact on the stability region, as illustrated on Figure 3. In particular, the condition (52) does not only involve λ 1 but also λ 2 . The strong stability region (Assumption 3.2 satisfied) in the plane (λ 1 , λ 2 ) for Q LW 2 corresponds to the grey and black disk, whereas the black region corresponds to the fulfillment of both (51) and (52).

Since the sequence (w j ) satisfies the induction relation (54), we have V n j = z n w j for all n ≤ N and j ≥ 1 -r. We apply Theorem 2.2 to V and obtain the following inequalities for all γ > 0: Using the expression of V and g for n ≤ N , we find: We choose γ > 0 such that |z| > exp(γ ∆t), then we let N tend to infinity, and the result follows.

Figure 1 .

 1 Figure 1. Lax-Friedrichs scheme (41): ratio between the norm of the trace and the norm of the source terms. Non-zero initial data and zero interior source term (left). Zero initial data and non-zero interior source term (right).

Figure 2 .

 2 Figure 2. Lax-Wendroff scheme (42): ratio between the norm of the trace and the norm of the source terms. Non-zero initial data and zero interior source term (left). Zero initial data and non-zero interior source term (right).

Figure 3 .

 3 Figure 3. Strong stability (grey and black) and coercivity (black) regions for the modified Lax-Wendroff scheme II in the plane (λ 1 , λ 2 ) for the specific example.

q+1 j=1 |w j | 2 ≤ C w 2 1 1 +

 221 -r,+∞ ∆t N n=0 (|z| e -γ ∆t ) 2 n -C 0 j=1-r |w j | 2 .

Research of the first author was supported by the Agence Nationale de la Recherche, contract ANR-08-JCJC-0132-01.

4.2.1. Lax-Friedrichs' scheme. The Lax-Friedrichs' scheme with homogeneous Dirichlet boundary condition is as follows:

with the operator Q LF given by:

In particular, it is of the form (32) with p 1 = p 2 = r 1 = r 2 = 1, q 1 = q 2 = 0, and obvious definitions for the matrices A j1,j2 . Assumptions 3.1 and 3.2 are translated in terms of admissible zones for the CFL parameters (λ 1 , λ 2 ).

Let us begin with Assumption 3.1. In this case, p 1 = 1, and the mapping:

has to be coercive on ℓ 2 (Z) for all |z| ≥ 1. Since A 1 is diagonalizable, a sufficient condition for Assumption 3.1 to be satisfied is:

(45) Assumption 3.2 is particularly simple in this case, because the symbol of the discretized operator Q LF is a normal matrix. Hence it is diagonalizable in an orthonormal basis and its Hermitian norm coincides with its spectral radius. We refer to [START_REF] Vaillancourt | On the stability of Friedrichs' scheme and the modified Lax-Wendroff scheme[END_REF] for the following sufficient condition for Assumption 3.2 to hold:

As shown in [START_REF] Turkel | Symmetric hyperbolic difference schemes and matrix problems[END_REF], the latter condition holds provided that we have 4 :

If conditions (45) and (46) hold, Theorem 3.2 shows that the solution to (44) satisfies the estimate of Theorem 3.2.

4.2.2. Modified Lax-Wendroff 's scheme I. Let us now address a variant of the Lax-Wendroff scheme introduced by Wendroff in [START_REF] Wendroff | Well-posed problems and stable difference operators[END_REF], whose ℓ 2 -stability has been further studied by Vaillancourt in [START_REF] Vaillancourt | On the stability of Friedrichs' scheme and the modified Lax-Wendroff scheme[END_REF].

With the notation of Section 3, the scheme is as follows:

2 ) . (47) For this scheme, p 1 = 2 and for all j 2 ∈ Z,

We first determine the CFL parameters (λ 1 , λ 2 ) for which Assumption 3.2 is satisfied. The symbol of the discretized operator Q LW 1 is given by:

where J(ξ) := sin ξ 1 λ 1 A 1 + sin ξ 2 λ 2 A 2 , and c(ξ) := (cos ξ 1 + cos ξ 2 )/2. In particular, G(ξ) is a normal matrix. Hence, G(ξ) is diagonalizable in an orthonormal basis and the scheme is ℓ 2 -stable if and only if Assumption 3.2 is satisfied. This stability condition is equivalent to the fulfillment of the von Neumann condition, which reads G(ξ) G * (ξ) ≤ I, that is:

4 Condition ( 46) is sufficient for stability, but it is also necessary when the matrices A 1 and A 2 commute.

Following [START_REF] Vaillancourt | On the stability of Friedrichs' scheme and the modified Lax-Wendroff scheme[END_REF], one notes that

Hence, Assumption 3.2 holds provided

This condition is equivalent to

Therefore, Q LW 1 satisfies Assumption 3.2 provided

Let us now turn to Assumption 3.1. In this case, for all z ∈ C * , we have:

and A 2 (z) is nothing but a simple multiplication. The mapping A 2 (z) is coercive if:

Consequently Theorem 3.2 shows stability for the modified Lax-Wendroff scheme (47) with Dirichlet boundary conditions provided ( 48) and (49) hold.

4.2.3. Modified Lax-Wendroff 's scheme II. Our last example is another modification of the Lax-Wendroff scheme introduced by Lax and Wendroff in [START_REF] Lax | Difference schemes for hyperbolic equations with high order of accuracy[END_REF]. The discretized operator is as follows:

Although it may seem at first glance that the scheme involves 5 points in each direction, there are cancellations for the terms T 2 j , T -2 j and we have r 1 = r 2 = p 1 = p 2 = 1. The ℓ 2 -stablity of Q LW 2 has been characterized by Lax and Wendroff in [START_REF] Lax | Difference schemes for hyperbolic equations with high order of accuracy[END_REF] (see also Turkel [START_REF] Turkel | Symmetric hyperbolic difference schemes and matrix problems[END_REF]). However there is a small gap between the general concept of ℓ 2 -stablity for which the operators Q n LW 2 are bounded uniformly in n ∈ N, and Assumption 3.2 where we require the norm of Q LW 2 to be not greater than 1. The latter property is called strong ℓ 2 -stability by Tadmor in [START_REF] Tadmor | Complex symmetric matrices with strongly stable iterates[END_REF]. The results of [START_REF] Lax | Difference schemes for hyperbolic equations with high order of accuracy[END_REF] and [START_REF] Tadmor | Complex symmetric matrices with strongly stable iterates[END_REF] show that the discretized operator Q LW 2 in (50) is ℓ 2 -stable if:

while Assumption 3.2 is satisfied under the slightly more restrictive condition:

Let us now address Assumption 3.1. Unlike the previous examples, A 1 (z) is not a multiplication anylonger. Here, for all z ∈ C * , we have:

I) w , and Assumption 3.1 is satisfied if and only if A 1 (1/2) is coercive. The symbol of A 1 (1/2) is given by:

Using Plancherel's Theorem, it is rather easy to prove that the operator A 1 (1/2) is coercive if and only if its symbol is invertible for all ξ 2 .

Appendix A. Proof of Lemma 2.5

Let us recall that the matrices A ℓ 's are defined in [START_REF] Kreiss | Initial boundary value problems for hyperbolic systems[END_REF]. Using Assumption 2.1, we can define the following matrix M(z) that depends holomorphically on z, with |z| ≥ 1:

(53) We recall the following result that can be found in [4, page 659]:

Lemma A. 1 ([4]). Let Assumptions 2.1 and 2.2 be satisfied. Then for all z ∈ C with |z| > 1, the matrix M(z) defined by (53) has no eigenvalue on the unit circle S 1 . We let E s (z) denote the generalized eigenspace associated with those eigenvalues of M(z) that belong to the unit disk.

The vector space E s (z) is made of all vectors of the form (w p , . . . , w 1-r ), where each w j belongs to C D , and such that the sequence (w j ) j≥1-r defined by:

belongs to ℓ 2 (the sequence is even exponentially decreasing). Observe that (54) defines the sequence (w j ) j≥1-r in a unique way thanks to Assumption 2.1. The proof of Lemma 2.5 relies on the following preliminary result that is entirely similar to the analysis in [1, page 110]:

Lemma A.2. Let Assumptions 2.1 and 2.2 be satisfied. Then there exists a constant C > 0 such that for all z ∈ C with |z| > 1, and for all W = (w p , . . . , w 1-r ) ∈ E s (z), the sequence (w j ) j≥1-r defined by (54) satisfies:

Proof of Lemma A.2. Let N ∈ N, and let z ∈ C with |z| > 1. We consider an element W = (w p , . . . , w 1-r ) of E s (z) and the sequence (w j ) j≥1-r ∈ ℓ 2 defined by (54). Then we define the source terms g by:

We let V denote the solution to the numerical scheme:

Let us now go back to the proof of Lemma 2.5. We consider the solution W to (26), and introduce the piecewise constant functions:

The definition of g j (t) is similar. We apply the Laplace transform to (26) and obtain:

where z is a short notation for exp((γ +i ω) ∆t), with γ +i ω the point where we evaluate the Laplace transform. Using Theorem 4.2 in [START_REF] Gustafsson | Stability theory of difference approximations for mixed initial boundary value problems[END_REF], we know that ( W j ) j≥1-r satisfies the estimate 5 :

with a constant C that is uniform with respect to z.

Since we have ( W j ) j≥1-r ∈ ℓ 2 , we know that ( W p , . . . , W 1-r ) ∈ E s (z). Consequently, we can apply Lemma A.2 and obtain:

where we use (56) to derive the last inequality. The conclusion of Lemma 2.5 follows after integrating (57) with respect to ω ∈ R and using Plancherel's Theorem. The proof of Lemma 2.5 is now complete. Let us observe that the estimate (56), that is obtained by using the strong stability of the discretization (3), is not sufficient to derive (57) since the control of the "interior" terms W 1 , . . . , W q+1 deteriorates as |z| tends to 1. The key point is to combine (56) with the estimate of Lemma A.2. The latter is a typical estimate in the frequency variables when the uniform Kreiss-Lopatinskii condition holds, see e.g. [1, chapter 4]. A major advantage of our approach is to bypass the verification of the uniform Kreiss-Lopatinskii condition for [START_REF] Michelson | Stability theory of difference approximations for multidimensional initial-boundary value problems[END_REF] in the frequency variables (this was the approach followed in [START_REF] Goldberg | Scheme-independent stability criteria for difference approximations of hyperbolic initialboundary value problems[END_REF] and it seems difficult to carry out for systems). Lemma A.2 appears here a direct and easy consequence of the energy method.