J E ´re

´my Jabiol
email: jeremy.jabiol@gmail.com

Eric Chauvet

Je ´re ´my Jabiol

Fungi are involved in the effects of litter mixtures on consumption by shredders

Keywords: aquatic hyphomycetes, biodiversity effect, decomposition, Gammarus fossarum, litter mixture

1. Decomposition of litter mixtures in both terrestrial and aquatic ecosystems often shows nonadditive diversity effects on decomposition rate, generally interpreted in streams as a result of the feeding activity of macroinvertebrates. The extent to which fungal assemblages on mixed litter may influence consumption by macroinvertebrates remains unknown. 2. We assessed the effect of litter mixing on all possible three-species combinations drawn from four tree species (Alnus glutinosa, Betula pendula, Juglans regia and Quercus robur) on both fungal assemblages and the rate of litter consumption by a common shredder, Gammarus fossarum. After a 9-week inoculation in a stream, batches of leaf discs were taken from all leaf species within litter mixture combinations. Ergosterol, an indicator of fungal biomass, and the composition of fungal assemblages, assessed from the conidia released, were determined, and incubated litter offered to G. fossarum in a laboratory-feeding experiment.

3. Mixing leaf litter species enhanced both the Simpson's index of the fungal assemblage and the consumption of litter by G. fossarum, but had no clear effect on mycelial biomass. Specifically, consumption rates of J. regia were consistently higher for mixed-species litter packs than for single-species litter. In contrast, the consumption rates of B. pendula were not affected by litter mixing, because of the occurrence of both positive and negative litter-mixing effects in different litter species combinations that counteracted each other. 4. In some litter combinations, the greater development of some fungal species (e.g. Clavariopsis aquatica) as shown by higher sporulation rates coincided with increased leaf consumption, which may have resulted from feeding preferences by G. fossarum for these fungi. 5. Where litter mixture effects on decomposition rate are mediated via shredder feeding, this could be due to indirect effects of the fungal assemblage.

Introduction

Ecosystems worldwide are currently suffering a dramatic rate of species extinctions [START_REF] Butchart | Global biodiversity: indicators of recent declines[END_REF]. Over the past few decades, a focus in ecology has been to assess the consequences of such a biodiversity loss on ecosystem processes [START_REF] Loreau | Biodiversity and Ecosystem Functioning: Synthesis and Perspectives[END_REF][START_REF] Naeem | Biodiversity, Ecosystem Functioning, and Human Wellbeing: An Ecological and Economic Perspective[END_REF]. While focussing first on the relationships between plant diversity and primary productivity [START_REF] Naeem | Declining biodiversity can alter the performance of ecosystems[END_REF][START_REF] Tilman | Productivity and sustainability influenced by biodiversity in grassland ecosystems[END_REF], research has more recently been extended to a wide variety of ecosystems and processes [START_REF] Ha ¨ttenschwiler | Biodiversity and litter decomposition in terrestrial ecosystems[END_REF][START_REF] Lecerf | Riparian plant species loss alters trophic dynamics in detritus-based stream ecosystems[END_REF][START_REF] Gessner | Diversity meets decomposition[END_REF]. For instance, many studies have been dedicated to the consequences of biodiversity loss at various trophic levels (e.g. litter, micro-and macrodecomposers) on leaf litter decomposition [START_REF] Lecerf | Biodiversity-ecosystem function research: insights gained from streams[END_REF][START_REF] Gessner | Diversity meets decomposition[END_REF][START_REF] Kominoski | Beyond species richness: expanding biodiversity-ecosystem functioning theory in detritus-based systems[END_REF], which is a key process for carbon and nutrient cycling in forested ecosystems (including forest soils and streams) [START_REF] Wallace | Multiple trophic levels of a forest stream linked to terrestrial litter inputs[END_REF][START_REF] Cebrian | Patterns in the fate of production in plant communities[END_REF]. A number of studies have shown that litter decomposition rates are predictable from traits of the plants and leaves concerned. These traits include those that make litter decomposition slower (e.g. lignin and tannin content, and cuticle toughness) or faster (e.g. a high content of nitrogen and phosphorus) [START_REF] Melillo | Nitrogen and lignin control of hardwood leaf litter decomposition dynamics[END_REF][START_REF] Webster | Vascular plant breakdown in freshwater ecosystems[END_REF][START_REF] Enriquez | Patterns in decomposition rates among photosynthetic organisms: the importance of detritus C : N : P content[END_REF][START_REF] Ostrofsky | Relationship between chemical characteristics of autumn-shed leaves and aquatic processing rates[END_REF]. Such traits may affect both the density (e.g. recruitment and colonisation success) and processing efficiency of decomposer organisms, mostly consisting of fungi and leaf-eating macroinvertebrates (i.e. shredders) in aquatic environments [START_REF] Cummins | Shredders and riparian vegetation[END_REF][START_REF] Baldy | Bacteria, fungi and the breakdown of leaf litter in a large river[END_REF][START_REF] Grac | The role of invertebrates on leaf litter decomposition in streams -A review[END_REF]Swan & Palmer, 2006a;Kominoski et al., 2009).

Patches of litter in streams are generally composed of several leaf species, however, and recent studies have emphasised how such mixing can influence decomposition rate and nutrient flux [START_REF] Srivastava | Diversity has stronger top-down than bottom-up effects on decomposition[END_REF][START_REF] Gessner | Diversity meets decomposition[END_REF][START_REF] Kominoski | Beyond species richness: expanding biodiversity-ecosystem functioning theory in detritus-based systems[END_REF][START_REF] Lecerf | Incubation time, functional litter diversity, and habitat characteristics predict litter mixing effects on decomposition[END_REF]. The decomposition rate of mixtures often diverges from the simple average of the species, that is, it is non-additive [START_REF] Ha ¨ttenschwiler | Biodiversity and litter decomposition in terrestrial ecosystems[END_REF][START_REF] Lecerf | Decomposition of diverse litter mixtures in streams[END_REF]. Both negative and positive effects of litter mixing on decomposition rate have been reported [START_REF] Mcarthur | Interspecific leaf interactions during decomposition in aquatic and floodplain ecosystems[END_REF]Swan & Palmer, 2006b;[START_REF] Swan | The role of native riparian tree species in decomposition of invasive tree of heaven (Ailanthus altissima) leaf litter in an urban stream[END_REF], and several non-exclusive mechanisms have been proposed to explain these mixture effects in both terrestrial and aquatic environments [START_REF] Ha ¨ttenschwiler | Biodiversity and litter decomposition in terrestrial ecosystems[END_REF][START_REF] Kominoski | Beyond species richness: expanding biodiversity-ecosystem functioning theory in detritus-based systems[END_REF]. The latter may include some transfers of nutrient or refractory compounds between leaves with different chemical properties [START_REF] Ha ¨ttenschwiler | Biodiversity and litter decomposition in terrestrial ecosystems[END_REF][START_REF] Schimel | Nitrogen transfer between decomposing leaves of different N status[END_REF], or alterations of decomposer community structure (Kominoski & Pringle, 2009) and feeding activity or behaviour (Swan & Palmer, 2006b;[START_REF] Sanpera-Calbet | Leaf diversity influences in-stream litter decomposition through effects on shredders[END_REF]. However, the relevance and relative strength of such mechanisms across ecosystems are still uncertain [START_REF] Gessner | Diversity meets decomposition[END_REF]. This lack of understanding makes the decomposition of mixtures, and thus the consequences of litter diversity loss (e.g. through species invasions, harvesting, forestry), often unpredictable from the attributes of the litter species pool.

In streams, the activity of shredders has often been suggested to be responsible for litter-mixing effects (Swan & Palmer, 2006a;[START_REF] Sanpera-Calbet | Leaf diversity influences in-stream litter decomposition through effects on shredders[END_REF]. For example, the increased decomposition of fast-decomposing species within mixtures can be explained by shredders first consuming litter species with the highest concentrations of nutrients (Swan & Palmer, 2006b), whereas the preference for refractory leaves by caddis larvae for case construction could potentially lead to increased decomposition of slowly decomposing species [START_REF] Kochi | Green leaves enhance the growth and development of a stream macroinvertebrate shredder when senescent leaves are available[END_REF][START_REF] Sanpera-Calbet | Leaf diversity influences in-stream litter decomposition through effects on shredders[END_REF]. It has been suggested that microbial activity alone does not alter the decomposition of litter mixtures [START_REF] Scha ¨dler | Do invertebrate decomposers affect the disappearance rate of litter mixtures?[END_REF]Swan & Palmer, 2006a;[START_REF] Schindler | Functional leaf traits and biodiversity effects on litter decomposition in a stream[END_REF]. The field studies assessing the relative contribution of micro-versus macrodecomposers on litter-mixing effects often rely on the use of coarse and fine mesh litter bags (Ba ¨rlocher, 2005a), assuming that the microbial contribution to decomposition corresponds to the mass loss observed in fine mesh bags, while the macroinvertebrates contribution is inferred from the difference between coarse and fine mesh bags. However, if fine mesh bags adequately assess microbial activity, the processes occurring in coarse mesh bags are not only the result of macroinvertebrate activity and microbial one, but also that of complex interactions occurring between macroinvertebrates and microorganisms such as bacteria and microalgae [START_REF] Franken | Growth of shredders on leaf litter biofilms: the effect of light intensity[END_REF], and fungi [START_REF] Lecerf | Riparian plant species loss alters trophic dynamics in detritus-based stream ecosystems[END_REF]. Consequently, the contribution of macroinvertebrate activity and these interactions cannot be disentangled using such an approach, which may result in misleading interpretations. Although the contribution of fungi is predominant for leaf litter decomposition and shredder feeding [START_REF] Baldy | Bacteria, fungi and the breakdown of leaf litter in a large river[END_REF][START_REF] Hieber | Contribution of stream detrivores, fungi, and bacteria to leaf breakdown based on biomass estimates[END_REF], the structure of fungal assemblages has rarely been assessed in litter mixture experiments, and its contribution to mixture effects on decomposition (through the interactions of fungi with shredders) remains unknown. Kominoski et al. (2009) suggested that litter mixing could influence microbial activity and diversity. For instance, the presence of recalcitrant litter could provide higher structural complexity to the leaf pack and allow a better circulation of oxygen, nutrients and conidia between leaves of different species. Leachates from different litter species could also stimulate (e.g. by leaching of nutrients; [START_REF] Tukey | The leaching of substances from plants[END_REF][START_REF] Jensen | Decomposition of angiosperm tree leaf litter[END_REF] or inhibit (e.g. by leaching of polyphenols; [START_REF] Suberkropp | Changes in the chemical composition of leaves during processing in a woodland stream[END_REF] fungal growth on other leaves. Finally, fungi were shown to have some substrate preferences, with litter-associated communities being dissimilar between different litter species [START_REF] Gulis | Are there any substrate preferences in aquatic hyphomycetes[END_REF]. Thus, diverse leaf packs might produce a more diverse pool of conidia (released from the various leaf species), with each litter species within mixtures being more likely to be colonised by a greater diversity of fungi than in single-species litter packs.

Such alterations in both biomass and diversity of fungi associated with leaf litter are expected to influence shredder feeding [START_REF] Arsuffi | Effects of fungal mycelia and enzymatically degraded leaves on feeding and performance of caddisfly (Trichoptera) larvae[END_REF][START_REF] Grac | Comparative ecology of Gammarus pulex and Asellus aquaticus (L.) II: fungal preferences[END_REF][START_REF] Lecerf | Riparian plant species loss alters trophic dynamics in detritus-based stream ecosystems[END_REF]. Actually, fungal activity leads to increased litter quality, favouring the activity of shredders that feed on both leaf tissues and fungal mycelium [START_REF] Ba ¨rlocher | Assimilation efficiency of Gammarus pseudolimnaeus (Amphipoda) feeding on fungal mycelium or autumn-shed leaves[END_REF][START_REF] Arsuffi | Effects of fungal mycelia and enzymatically degraded leaves on feeding and performance of caddisfly (Trichoptera) larvae[END_REF][START_REF] Grac | Importance of fungi in the diet of Gammarus pulex and Asellus aquaticus II. Effects on growth, reproduction and physiology[END_REF] and exhibiting preferences for different fungal species [START_REF] Arsuffi | Selective feeding by stream caddisfly (Trichoptera) detritivores on leaves with fungal-colonized patches[END_REF][START_REF] Grac | Comparative ecology of Gammarus pulex and Asellus aquaticus (L.) II: fungal preferences[END_REF]. Thus, fungal diversity may alter shredder feeding through two mechanisms: (i) complementarity in resource use, leading to a better conditioning of the litter (i.e. enzymatic complementarity) and higher total fungal biomass; and (ii) complementarity as a nutritional resource, resulting in enhanced activity of shredders [START_REF] Lecerf | Riparian plant species loss alters trophic dynamics in detritus-based stream ecosystems[END_REF][START_REF] Duarte | Aquatic hyphomycete diversity and identity affect leaf litter decomposition in microcosms[END_REF].

We tested the hypothesis that litter mixture effects on decomposition result from (i) the alteration of fungal biomass and assemblage structure at the local (i.e. leaf) scale; (ii) which in turn influences the activity of shredders. In a woodland stream, we placed leaf litter from four contrasting species in fine mesh bags and subjected them to both single-species and three-species mixture treatments, thus allowing us to evaluate mixing effects in a statistically balanced design (i.e. four single-species versus four mixed treatments). Then, we compared fungal assemblages, mycelial biomass and the consumption efficiency by a common detritivore, Gammarus fossarum (Koch, 1836), on individual litter species across the different treatments.

Methods

Experimental design

Leaf litter was exposed in the Re ´millasse ´, an oligotrophic second-order stream in the French Pyrenees (01°05¢24¢¢E; 42°56¢36¢¢N, 480 m a.s.l). Four tree species (alder, Alnus glutinosa (L.) Gaertn.; birch, Betula pendula Roth; walnut, Juglans regia L.; and oak, Quercus robur L.) were used. The leaves of these species differ in traits, such as the content of nutrients [e.g. nitrogen (N) and phosphorus (P)] and structural compounds (e.g. cellulose and lignin) that influence the decomposition process, it being fastest for alder and slowest for oak, with the remaining species being intermediate [START_REF] Webster | Vascular plant breakdown in freshwater ecosystems[END_REF][START_REF] Ostrofsky | Relationship between chemical characteristics of autumn-shed leaves and aquatic processing rates[END_REF][START_REF] Leroy | Litter quality, stream characteristics and litter diversity influence decomposition rates and macroinvertebrates[END_REF]. We determined these parameters on three leaf batches for each leaf species after 24 h of leaching in tap water (Table 1).

We used four replicates each for all four single-species and all four three-species treatments, resulting in a total of 32 litter bags. Litter bags were made of fine mesh (0.3 mm), thus preventing the access of macroinvertebrates, and were introduced in the stream in late autumn (16 November 2009). Each litter bag contained a total of 6 g (i.e. 2 g per species for mixtures) of litter dried at ambient temperature. Litter bags were removed from the stream after 9 weeks of exposure, and the leaves cleaned and separated by species. For each litter bag, three batches of ten 10-mm leaf discs were cut (i.e. one batch per species in mixtures and three batches of the same species for single-species litter bags), except for the litter mixtures containing alder, oak and walnut in which alder leaves were too far decomposed to cut discs in three of the four replicates, resulting in a total of 93 batches (i.e. 930 leaf discs).

Fungal communities

All leaf discs were placed into glass Petri dishes containing 20 mL of filtered (GF ⁄ C glass fibre filter, 1.2 lm pore size; Whatman, Clifton, NJ, U.S.A.) stream water and kept at 10°C during 48 h under constant agitation (100 rpm). The water containing released conidia was then preserved with 2% formalin (final concentration). To characterise fungal assemblages, an aliquot of these conidial suspensions was filtered on a membrane filter (SMWP, 5 lm porosity; Millipore, Bedford, MA, U.S.A.) and stained with Trypan blue (0.1% in 60% lactic acid). Trapped conidia were then counted and aquatic hyphomycete species identified under the microscope at •200-400 (see Ba ¨rlocher, 2005b). The Simpson's dominance index was determined from conidial numbers and used as a measure of fungal assemblage structure.

Fungal biomass was assessed through determination of ergosterol by HPLC on four (of the 10 per batch) leaf discs that were stored frozen, then freeze-dried and weighed to the nearest 0.01 mg [see [START_REF] Gessner | Use of solid-phase extraction to determine ergosterol concentrations in plant tissue colonized by fungi[END_REF] and [START_REF] Lecerf | Riparian plant species loss alters trophic dynamics in detritus-based stream ecosystems[END_REF] for more details]. Mycelial biomass was calculated using species-specific conversion factors weighted by the proportions of conidia released by the different species in each sample. Species-specific conversion factors were available for 13 species (Table 2) from Gessner & Chauvet (1993, 1994) and [START_REF] Suberkropp | Comparison of ATP and ergosterol as indicators of fungal biomass associated with decomposing leaves in streams[END_REF], which comprised all dominant species, and accounted for 83% of total conidial production. The average conversion factor of 5.5 mg ergosterol per g of mycelium was used for the 14 remaining species (Gessner & Chauvet, 1993). Mycelial biomass was expressed as a percentage of total detrital dry mass.

Feeding experiment

Three of the six remaining leaf discs per batch were put into 5 • 6 • 4 cm containers filled with 20 mL of filtered (Whatman GF ⁄ C glass fibre filter) stream water. Each container also contained one individual of G. fossarum (3.56 mg ± 0.07 SE, n = 93). All individuals were collected in a nearby stream and acclimated 1 week in the laboratory, fed with natural litter collected in their stream of origin. They were then starved for 24 h before the beginning of the experiment. Specimens of G. fossarum were kept at 10°C and allowed to feed for 90, 114, 140 and 165 h on birch, walnut, alder and oak leaves, respectively. Then leaf discs were removed, cleaned, freeze-dried and weighed to the nearest 0.01 mg. Each sample was paired with a control for the microbially mediated decomposition occurring during the feeding experiment, consisting of three additional discs under the same conditions but without detritivores, which resulted in a total of 186 containers. The leaf mass loss because of invertebrates was determined as the difference of final leaf mass remaining between treatments where G. fossarum was absent and present. Then the invertebrates were oven-dried (60°C, 48 h) and weighed to the nearest 0.01 mg, and the consumption rates were calculated as the ratio between leaf consumption and invertebrate body mass and expressed in g leaf DM g )1 day )1 .

Statistical analyses

First, a non-metric multidimensional scaling was performed on fungal communities using the Bray-Curtis distance to illustrate the differences in composition between substrate species and inoculation contexts (i.e. litter diversity within litter pack).

Then the differences between treatments were tested by performing ANOVA ANOVAs to assess the effect of both litter identity and litter mixing on different target variables, including the consumption rate by G. fossarum, the fungal biomass and the Simpson's index of dominance in fungal assemblages. Litter pack identity was included as a random factor to control the fact that these variables were not independent from each other when evaluated on different batches from the same litter packs. As a second step, we performed the same analysis using a priori contrasts that allow direct comparison of litter-mixing effect on different litter species instead of overall mixing effect.

Finally, the effect of the species composition of the litter pack, rather than litter species richness, was assessed by comparing the consumption rates of the four species originating from litter packs of different species composition. Comparisons were performed using pairwise Wilcoxon nonparametric test with Bonferroni correction for multiple comparisons. Consumption rates and Simpson's dominance index in all analyses were log-transformed, while mycelial biomass was square-root transformed to meet normality and homoscedasticity assumptions (checked graphically). All statistics were performed using the R 2.6.0 software (The R Core Team, 2007).

Results

Fungal community structure

Different plant litter species supported different fungal assemblages (Fig. 1). In particular, communities on oak litter contained a large proportion of Clavariopsis aquatica, Clavatospora longibrachiata and Tricladium chaetocladium, the latter being in low abundance on other litter species (Table 2). Similarly, fungal assemblages associated with birch differed from the others because of their very high proportion of both Flagellospora curvula and C. longibrachiata, although these were present in substantial proportions on other litter species. Alatospora acuminata was more abundant on walnut than on the other litter species. Finally, assemblages associated with alder were intermediate, although closer to those on walnut than the two other species.

Litter mixing had little effect on fungal assemblages, especially when compared with the influence of leaf identity (Fig. 1). Nevertheless, the dissimilarity among the assemblages from different litter species tended to be higher for samples originating from mixed than singlespecies litter (except for those associated with walnut), corresponding to an increased abundance of species already dominant in single-species treatments. This coincided with a significant effect of litter mixing on Simpson's index of dominance (Table 3), which was higher on oak (0.26 ± 0.02 SE) and birch (0.22 ± 0.01 SE) than on alder and walnut (both: 0.14 ± 0.01 SE). Assemblages associated with samples from diverse litter packs were actually less even than those from single-species litter packs, with average Simpson's dominance indices of 0.21 (±0.01 SE) and 0.18 (±0.01 SE), respectively (Fig. 2). A net positive effect of litter mixing on fungal dominance was observed, and depended on litter pack composition (e.g. 38% more C. aquatica on oak when mixed with birch and walnut, and 48% more C. longibrachiata on birch when mixed with oak and walnut, than in other mixtures; data not shown), but was only significant on birch litter in the contrast analysis (Fig. 2).

Fungal biomass and consumption by G. fossarum

Litter species identity strongly affected the fungal biomass associated with leaves and consumption rates of G. fossarum (ANOVA ANOVA; Table 3). The mean contribution of mycelial biomass to total detrital mass ranged from Fig. 1 Non-metric multidimensional scaling of fungal assemblages associated with litter species from single-species (white dots) and mixed (grey dots) treatments. Bars are the standard errors associated with the mean of coordinates on axes 1 and 2. .4 mg g )1 (±6.2 SE) of total detrital mass, respectively. Fungal biomass did not differ significantly with litter pack composition for any litter species, although discrepancies between leaves of the same species originating from some litter mixtures occurred. For instance, mycelial biomass associated with walnut was low when the latter was mixed with alder and oak (76.1 mg g )1 ± 21.0 SE), but reached 124.3 mg g )1 (±13.2 SE) in litter packs containing both birch and oak (Fig. 3). Mycelial biomass on birch did not differ significantly (Table 3) when it was mixed with either alder and walnut or oak and walnut (170.5 mg g )1 ± 17.9 SE and 170.2 mg g )1 ± 31.8 SE, respectively) or in litter packs containing alder and oak (138.4 mg g )1 ± 16.6 SE). Consumption of oak by G. fossarum was slowest (0.16 g g )1 day )1 ± 0.03 SE), and the consumption of walnut (0.57 g g )1 day )1 ± 0.05) and birch (0.35 g g )1 day )1 ± 0.04) faster than that of alder (0.25 g g )1 day )1 ± 0.04). Moreover, litter mixing and species identity had an interactive effect on consumption rate (Table 3), meaning that the mixture effect on consumption depended on the litter species considered. Specifically, mixing litter led to higher consumption rate on walnut leaves by 31%, but did not significantly influence the consumption of other litter species (Fig. 2). For birch (and, to a lesser extent, oak), the overall non-significant effect of litter mixing on consumption by G. fossarum (Table 3) resulted from both positive and negative effects of certain litter species from mixtures (Fig. 3).

Species composition of litter mixtures

Discussion

In accordance with the literature, our results suggest that litter mixing can have both negative and positive effects on leaf litter decomposition [START_REF] Swan | Leaf diversity alters litter breakdown in a Piedmont stream[END_REF], fungal assemblages (Kominoski et al., 2009) and shredder processing ability (Swan & Palmer, 2006a), while litter diversity per se has no effect [START_REF] Wardle | Biodiversity and plant litter: experimental evidence which does not support the view that enhanced species richness improves ecosystem function[END_REF][START_REF] Scha ¨dler | Do invertebrate decomposers affect the disappearance rate of litter mixtures?[END_REF][START_REF] Lecerf | Decomposition of diverse litter mixtures in streams[END_REF][START_REF] Swan | Leaf litter species evenness influences nonadditive breakdown a headwater[END_REF]. This is an overall result of the counteracting effects of particular species in the various mixtures [START_REF] Srivastava | Diversity has stronger top-down than bottom-up effects on decomposition[END_REF]. This result has been consistently shown across ecosystems (e.g. terrestrial and aquatic) and may result from similar mechanisms, although their relative importance may be influenced by discrepancies in habitat conditions and in the functioning of decomposer communities in the various ecosystems [START_REF] Gessner | Diversity meets decomposition[END_REF].

Litter-mixing effects on decomposition have often been interpreted as a result of the activity of shredders. For instance, preferential feeding on labile litter (Swan & Palmer, 2006b), or indirect benefits of refractory species (e.g. providing structured habitat and shelter against predators; [START_REF] Ha ¨ttenschwiler | Biodiversity and litter decomposition in terrestrial ecosystems[END_REF][START_REF] Sanpera-Calbet | Leaf diversity influences in-stream litter decomposition through effects on shredders[END_REF], have been suggested to drive increased decomposition of labile litter within litter mixtures. Our experimental design allowed us to exclude such mechanisms and demonstrate that litter-mixing effects on decomposition can also involve microorganisms, interacting with their physical environment and shredders. Litter mixtures altered both the fungal assemblages associated with individual litter species and their palatability to G. fossarum, which supports the notion of an indirect fungal contribution to the effect of mixing litter on decomposition by detritivores.

The effects of litter mixing on fungal assemblages

For a given litter species, the structure of fungal assemblages from mixtures was less even (i.e. with increased fungal dominance) than that from monocultures, which partly contrasts with findings by Kominoski et al. (2009). Unlike Kominoski et al. (2009), who assessed microbial assemblages using molecular approaches, we relied on the species identity from the pool of conidia produced, which may reflect some modifications of fungal assemblages structure and ⁄ or alterations of the relative spore production in different fungal species. In our experiment, litter mixing favoured the sporulation of the already-dominant fungal species. The fact that distinct assemblages were observed on different substrates, even in mixtures where these were close together, suggests that the structure of the fungal assemblage is more constrained by litter quality than species composition of the pool of conidia likely to colonise. To some extent, litter quality may act as an environmental filter, benefiting some species in their establishment and development [START_REF] Keddy | Assembly and response rules: two goals for predictive ecology[END_REF][START_REF] Canhoto | Leaf barriers to fungal colonization and shredders (Tipula lateralis) consumption of decomposing Eucalyptus globulus[END_REF][START_REF] Gulis | Are there any substrate preferences in aquatic hyphomycetes[END_REF][START_REF] Dang | Influence of conidial traits and leaf structure on attachment success of aquatic hyphomycetes on leaf litter[END_REF], potentially leading to their dominance through competitive exclusion [START_REF] Hardin | The principle of competitive exclusion[END_REF][START_REF] Diamond | Assembly of species communities[END_REF][START_REF] Treton | Competitive interaction between two aquatic hyphomycete species and increase in leaf litter breakdown[END_REF]. In contrast to fungal assemblage structure, no clear evidence for an effect of litter mixing on mycelial biomass associated with a given litter species was found. Again, litter identity was the main factor controlling mycelial biomass, probably reflecting the differences in the decomposition stage and nutritional value of different leaf species at the end of the experiment.

Litter-mixing effects on litter palatability

Unsurprisingly, litter identity was the most important factor influencing the consumption by G. fossarum. These differences may reflect litter chemistry, with the higher consumption rates on walnut and birch, two species that are relatively rich in P, suggesting that this nutrient is limiting for G. fossarum [START_REF] Evans-White | Taxonomic and regional patterns in benthic macroinvertebrate elemental composition in streams[END_REF][START_REF] Hladyz | Resource quality and stoichiometric constraints on stream ecosystem functioning[END_REF]. Low consumption of oak litter, although rich in P, was probably due to the high concentration of structural compounds, such as lignin, which inhibit shredder feeding [START_REF] Ostrofsky | Relationship between chemical characteristics of autumn-shed leaves and aquatic processing rates[END_REF]. Finally, the low consumption of alder litter reflected its low N (<2%; Table 1) and high lignin (>20%; Table 1) content, relative to its usual concentration range in nature, both parameters being strongly correlated with the decomposability of alder leaves [START_REF] Lecerf | Intraspecific variability in leaf traits strongly affects alder leaf decomposition in a stream[END_REF].

Overall leaf consumption by G. fossarum was not affected by litter mixing. However, both positive and negative effects of litter pack composition were observed, but with no clear relationship with the fungal assemblage and biomass. For instance, the consumption of oak, on which the structure of the fungal assemblage was the most responsive to litter mixture, was weakly affected by litter mixing. In contrast, the consumption of walnut was enhanced in all mixtures, while that of birch was higher in litter mixtures with walnut and oak, but lower in the others. Although these responses of walnut and birch consumption to mixture were not clearly related to any effect on fungal biomass, the highest consumption rates were observed in samples in which mycelial biomass was also higher than average, supporting the involvement of fungi in such effects. Moreover, increased proportions of C. aquatica and C. longibrachiata on oak and birch (as found in some litter mixtures), coincided with the highest consumption rates observed on these litter species. This is consistent with the previous findings on C. aquatica, showing that Gammarus sp. exhibited high consumption rates on litter colonised by this species [START_REF] Arsuffi | Selective feeding by shredders on leaf-colonizing stream fungi: comparison of macroinvertebrate taxa[END_REF]. Our result would thus suggest that C. longibrachiata could also be a prime resource for G. fossarum. Other fungal species that are known to be preferred by Gammarus and observed in this study include A. acuminata, Heliscus lugdunensis and Anguillospora longissima [START_REF] Arsuffi | Selective feeding by shredders on leaf-colonizing stream fungi: comparison of macroinvertebrate taxa[END_REF][START_REF] Grac | Comparative ecology of Gammarus pulex and Asellus aquaticus (L.) II: fungal preferences[END_REF], but the proportion of these species was low and not affected by litter mixing. Some other dominant fungal species observed in this study are not preferred by Gammarus (e.g. F. curvula) [START_REF] Ba ¨rlocher | Fungi in the diet of Gammarus pseudolimnaeus (Amphipoda)[END_REF][START_REF] Arsuffi | Selective feeding by shredders on leaf-colonizing stream fungi: comparison of macroinvertebrate taxa[END_REF][START_REF] Grac | Comparative ecology of Gammarus pulex and Asellus aquaticus (L.) II: fungal preferences[END_REF]. Alterations in the fungal assemblage that result in increased proportions of such non-preferred fungal species may lead to no overall effect of litter mixing on shredder consumption, or even explain negative litter-mixing effects, if the unpalatable fungal species grow to the detriment of preferred ones, or if they include species rejected by consumers (e.g. those containing repellents). Finally, the fact that three of the four replicates of alder leaves from litter mixtures containing alder, oak and walnut were in an advanced state of decomposition suggests an increased microbial activity in this particular litter combination.

If litter mixing can influence microbial assemblages associated with a given leaf, how these alterations propagate to affect the consumption rate of shredders is hypothetical. Alterations of fungal assemblages do not necessarily lead to the changes in fungal biomass and ⁄ or leaf palatability. Conversely, the alteration of leaf palatability can occur independently of changes in the fungal assemblage, possibly implying the involvement of other mechanisms. For instance, it has been demonstrated that nutrient transfers between litter species occur in terrestrial ecosystems [START_REF] Schimel | Nitrogen transfer between decomposing leaves of different N status[END_REF], and thus could be an important factor in determining the decomposition rate of mixtures (see [START_REF] Gessner | Diversity meets decomposition[END_REF]. By modifying nutrient balance in litter, such transfers are expected to alter fungal assemblages and shredder consumption rates, potentially to a different extent depending of fungal versus shredder respective needs. For instance, Gu ¨sewell & Gessner (2009) found that the N : P ratios in litter influenced colonisation by fungi, the latter being N-limited at low N : P ratios but becoming P limited at higher N : P ratios, leading to maximal microbial biomass at intermediate N : P supply ratios. On the other hand, Gammarus has a low N : P ratio [START_REF] Evans-White | Taxonomic and regional patterns in benthic macroinvertebrate elemental composition in streams[END_REF][START_REF] Hladyz | Resource quality and stoichiometric constraints on stream ecosystem functioning[END_REF] and thus may remain P-limited whichever litter it feeds on. Thus, Gammarus and fungi may be limited by different nutrients depending on the N : P ratios of their shared resource, leading to differences in their responses to the changes in litter composition following nutrient transfer. Initial litter nutrient contents, in relation to the respective needs of decomposers, may explain why nutrient transfer (when it occurs) may have positive or no effect on fungal-and shredder-induced decomposition. However, such an interpretation remains a matter of conjecture since other compounds, such as fatty acids and vitamins, are important in determining the activity of both shredders and microbial decomposers.

In conclusion, these results suggest that further investigation of microbial decomposers and nutrient transfers are needed to understand and predict the consequences of riparian diversity loss on stream ecosystem processes. Until now, litter-mixing effects on decomposition have been suggested to be attributable to altered shredder densities and diversity (Kominoski & Pringle, 2009), preferential feeding and complementary acquisition of nutrients from different leaf species (Swan & Palmer, 2006b) and increased habitat complexity in litter packs containing refractory litter [START_REF] Sanpera-Calbet | Leaf diversity influences in-stream litter decomposition through effects on shredders[END_REF]. In this experiment, litter-mixing effects on shredder feeding rates were observed without these proposed mechanisms. In contrast, our results suggest that aquatic hyphomycetes per se have the potential to explain mixture effects on litter decomposition by shredders. Because shredders show preferences for fungal species [START_REF] Arsuffi | Selective feeding by stream caddisfly (Trichoptera) detritivores on leaves with fungal-colonized patches[END_REF], any alteration of the structure of fungal assemblage associated with a litter species may influence shredder feeding rates, depending on the identity of the fungi (preferred or not) whose abundance is altered. Moreover, at larger spatiotemporal scales, changes in the species composition and diversity of riparian vegetation may lead to longer term alterations of aquatic hyphomycetes (Ba ¨rlocher & [START_REF] Ba ¨rlocher | Exotic riparian vegetation lowers fungal diversity but not leaf decomposition in Portuguese streams[END_REF][START_REF] Laitung | Vegetation diversity increases species richness of leaf-decaying fungal communities in woodland streams[END_REF]; but see [START_REF] Kominoski | Riparian forest composition affects stream litter decomposition despite similar microbial and invertebrate communities[END_REF] and shredder assemblages [START_REF] Whiles | Leaf litter decomposition and macroinvertebrate communities in headwater streams draining pine and hardwood catchments[END_REF][START_REF] Mckie | Assessing ecosystem functioning in streams affected by forest management: increased leaf decomposition occurs without changes to the composition of benthic assemblages[END_REF]. In nature, where shredders can be diverse, concomitant alterations of fungal and shredder assemblages following the changes in litter diversity may alter resource partitioning among shredder species exhibiting different fungal preferences. This could amplify the effect of the mechanism described in this study, by which alterations in litter diversity may have ramifying the effects on fungal assemblages and litter decomposition. Whether this mechanism is more significant than nutrient transfer among different leaf species, another mechanism possibly involving fungi (see [START_REF] Mctiernan | Respiration and nutrient release from tree litter mixtures[END_REF][START_REF] Schimel | Nitrogen transfer between decomposing leaves of different N status[END_REF], cannot be determined from our experiment. It remains possible, however, that the alteration in shredder feeding caused by litter mixing found in other studies (Swan & Palmer, 2006b;[START_REF] Sanpera-Calbet | Leaf diversity influences in-stream litter decomposition through effects on shredders[END_REF] could be due to changes in the fungal assemblage, as observed on some litter species in the present work.

Fig. 2

 2 Fig. 2 Average (±SE, n = 12 per bar) consumption rate, mycelial biomass and Simpson's dominance index of leaf-associated fungal communities for each litter species within single-species (white bars) and mixtures treatments (grey bars). A, alder; B, birch; W, walnut; and O, oak. Stars indicate litter species in which mixing effect was significant (a priori contrasts).

Fig. 3

 3 Fig. 3 Average consumption rate (grey bars) and mycelial biomass (black bars) (±SE, n = 4 per bar) for the four litter species contained in the three-species mixtures. The composition of mixtures is indicated by the first initial of the litter species' common name. Stars indicate the consumption values that are statistically different between mixture and single-species treatments for the same litter species (Wilcoxon nonparametric test).

Table 1

 1 Phosphorus, nitrogen, lignin and cellulose content of leaf species (mean ± SD; n = 3)

		P	N	Cellulose	Lignin
		(mg g )1 )	(mg g )1 )	(mg g )1 )	(mg g )1 )
	Alder	0.13 ± 0.02	18.59 ± 2.5	224.6 ± 6.4	225.5 ± 4.4
	Birch	0.29 ± 0.02	4.94 ± 0.5	195.3 ± 12.0	254.1 ± 15.9
	Walnut	0.17 ± 0.03	10.31 ± 0.1	231.3 ± 17.7	286.1 ± 8.8
	Oak	0.61 ± 0.13	9.21 ± 1.4	262.9 ± 14.6	313.4 ± 14.1

Table 2

 2 Aquatic hyphomycete species on each leaf litter species. Symbols represent the average proportion of conidia belonging to the species considered (results from single-species treatment and all mixtures combined; +++ >20%, ++ 10-20%, + 5-10% and AE <5%)

	Alder	Birch	Walnut	Oak

*Denotes species for which specific conversion factors were available from the literature and used to calculate mycelial biomass from ergosterol content (see Methods).

Acknowledgments

We thank Jean Clobert, Head of the Station d'Ecologie Expe ´rimentale du CNRS a `Moulis, for providing facilities, Andre ´Frainer, Sylvain Lamothe and Didier Lambrigot for technical assistance as well as Mark O. Gessner, Edwin T. H. M. Peeters, Alan G. Hildrew and two anonymous referees for their constructive comments on this manuscript. We also thank Richard Illi and Eawag's AUA Lab, Andreas Bruder and Andre ´Frainer for nutrient and lignin analyses. This research was funded through the CNRS and the European Science Foundation's (ESF) EURODI-VERSITY programme, which supported BioCycle as a collaborative research project. BioCycle has endorsed by DIVERSITAS as contributing to its biodiversity research priorities.


Philippe Chartier
email: philippe.chartier@inria.fr

Florian Méhats
email: florian.mehats@univ-rennes1.fr

Mechthild Thalhammer
email: mechthild.thalhammer@uibk.ac.at

Yong Zhang

Superconvergence Of Strang

Superconvergence of Strang splitting for NLS in T d

Keywords: highly-oscillatory systems, averaging, partial differential equation, Schrödinger equation, nonlinear, Strang splitting, splitting schemes. MSC numbers: 34K33, 37L05, 35Q55

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In this work, we analyze the convergence of Strang splitting method for nonlinear Schrödinger equations of the form

i∂ t u ε = - 1 ε ∆u ε + f (|u ε | 2 )u ε , u ε (0) = u 0 ∈ H σ (T d ), t ≤ T, (1)
where ∆ is the Laplacian operator acting on the Hilbert space H σ of functions on T d = [0, 2π] d with derivatives up to order σ and f is a C ∞ -function from R into itself such that f (0) = 0. 1 The operator ∆ being self-adjoint, Stone's theorem asserts that it generates a strongly continuous one-parameter unitary group e iτ ∆ which, here, is in addition 1-periodic in the variable τ = t/ε, so that on a fixed interval of time, the number of oscillations tend to +∞ as ε goes to zero, thus making the problem highly-oscillatory. It is customary for the ease of analysis to reparametrize the time and consider the evolution in the variable τ . In this language (but denoting again by t the new variable τ ), the equation reads

i∂ t u ε = -∆u ε + εf (|u ε | 2 )u ε , u ε (0) = u 0 ∈ H σ (T d ), t ≤ T /ε, (2)
the period of the group e it∆ is 1, and the equation must be solved on the interval [0, T /ε] (where interesting dynamics of the solutions appear). Of course, the number of oscillations ⌊T /ε⌋ over the interval of resolution still goes to ∞ for vanishing ε. Albeit the occurrence of ε in front of the nonlinearity, we insist that (2) is not a perturbation problem, owing to the length of the interval of time which scales like 1/ε. In the sequel, we give convergence results for semi-discretized numerical schemes applied to [START_REF] Bao | Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime[END_REF], keeping in mind that they are straightforwardly transferred to the original format (as stated in the Abstract). Accordingly, we assume in the whole paper the following:

Hypothesis (H) There exist T > 0 and ε 0 > 0, such that, for any 0 < ε < ε 0 , equation (2) has a unique solution u ε (t, •) ∈ C 1 ([0, T /ε]; H σ (T d )) with σ > d/2 + 2 and σ ≥ 4. Furthermore, there exists a positive constant K such that

∀0 < ε < ε 0 , ∀0 ≤ t ≤ T /ε, u ε (t) H σ ≤ K u 0 H σ . (3)
For convenience, we shall denote in the sequel, R = 2K u 0 H σ and B s ρ = {u ∈ H s , u H s ≤ ρ}, so that under previous hypothesis, the solution u ε (t) ≡ u ε (t, •) remains in B σ R/2 for all t ∈ [0, T /ε]. Note that in our context, the validity of this assumption can be rigorously established (see for instance [START_REF] Castella | Stroboscopic averaging for the nonlinear Schrdinger equation[END_REF]).

In this paper, we study the approximation properties of a semi-discretization in time. The numerical scheme that we consider is Strang splitting method, yielding approximations

Φ h (u) = ϕ h/2 T • ϕ h V • ϕ h/2 T (u),
where h is the step size and ϕ h T and ϕ h V are the exact flows of respectively

i∂ t u = -∆u and i∂ t u = εf (|u| 2 )u. ( 4
)
By Stone's theorem and the gauge invariance, they can be written for all u ∈ H s and all h ∈ R as

ϕ h T (u) = e ih∆ u and ϕ h V (u) = e -ihεf (|u| 2 ) u. Note that both maps are isometries of L 2 , that is, one has for all u ∈ L 2 , ϕ h T (u) L 2 = u L 2 and ϕ h V (u) L 2 = u L 2 .
It is perfectly clear that the first map is also an isometry in H s for all 0 ≤ s ≤ σ.

In agreement with the results of [START_REF] Chr | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF] derived for NLS in R 3 and f (u) = u, the sequence of approximations (Φ h ) n (u 0 ) are expected to converge on [0, T /ε] with second-order error estimates of the form

(Φ h ) n (u 0 ) -u(t n ) H σ-4 ≤ Const h 2
for t n = nh ≤ T /ε and sufficiently small h. Note that here σ and d are arbitrary with the only constraint that σ > d/2 + 2 and that the convergence is uniform in ε. The aim of this paper is to show that, somehow unexpectedly at first glance, this error estimate can be refined to obtain

(Φ h ) n (u 0 ) -u(t n ) H σ-2m ≤ Const (εh 2 + h m )
where 2 ≤ m ≤ ⌊σ/2⌋ depends on the smoothness of the initial value u 0 , provided the step size h is chosen in such a way that 1/h is an integer. 2 The precise formulation of this result is given in Theorem 5.5.

The paper is organized more or less along the following inclusions:

B σ R/2 exact solution   ⊂ B σ R ⊂ B σ-2 R f unctional bounds    ⊂ B σ-2 3R/4 h 1 -error, H s -stability ⊂ H σ-4 h 2 -error ⊂ H σ-2m h 2 ε-error
In Section 2, we consider the two functions u → F (u) = -if (|u| 2 )u and (τ, u) → F τ (u) = e -iτ ∆ F e iτ ∆ u which are essential ingredients of further proofs and derive bounds of their derivatives with respect to both u and τ . In addition, we give a Lipschitz estimate for F , which lies at the core of stability estimates derived in Section 3. In the spirit of [START_REF] Chr | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF] again, stability holds in H s -norms for 0 ≤ s ≤ σ -2 provided the numerical solutions remain in H σ-2 . Bounds on derivatives of F and F τ are then used in Section 4 to analyze the local truncation errors. In Section 5, we first briefly (re)-derive the (uniform in ε) convergence results of [START_REF] Chr | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF] (though without resorting to Lie-derivatives) and then analyze in great details the accumulation of errors in the case where 1/h ∈ N. The final result is then obtained in two steps, by first studying the errors over one period (see Theorem 5.4) where an averaging effect essentially kills the main error term and then extending the so-obtained estimates to the whole interval (see Theorem 5.5) . Finally, Section 7 presents numerical experiments confirming the occurrence of an additional ε-factor in error bounds. A few more technical results are exposed in Appendix.

2 Bounds on F , F τ and their derivatives

The embeddings of H σ and H σ-2 into L ∞ implies that H σ and H σ-2 are algebras: there exists a constant A > 1 such that for s = σ and s = σ -2, one has

∀(u, v) ∈ H s × H s , uv H s ≤ A u H s v H s .
In this context, any C ∞ -function G from C to itself, such that G(0) = 0, satisfies a so-called tame estimate (see [START_REF] Alinhac | Pseudo-differential operators and Nash-Moser theorem[END_REF]): there exists a non-decreasing function χ G from R + into R + , such that for s = σ and s = σ -2, one has

∀u ∈ H s , G(u) H s ≤ χ G ( u L ∞ ) u H s ≤ χ G (c u H s ) u H s , (5)
where we have used in the second part of the inequalities the continuous embedding of H σ-2 and H σ into L ∞ and denoted c the corresponding constant

• L ∞ ≤ c • H σ-2 ≤ c • H σ .
More generally, when G(0) is not assumed to vanish, we may consider G(u) = G(u) -G(0) and assert that there exists again a non-decreasing function χ G from R + into R + , such that for s = σ and s = σ -2, one has

∀u ∈ H s , G(u) H s ≤ (2π) d/2 |G(0)| + χ G ( u L ∞ ) u H s . (6)
In the sequel, the following functions will play an important role:

F : H σ → H σ u → -if (|u| 2 )u
and

F τ : [0, 1] × H σ → H σ (τ, u) → e -iτ ∆ F (e iτ ∆ u).
Let us notice that F ≡ F 0 , so that we will mostly concentrate on estimates for F τ .

A Lipschitz inequality for F τ . Regarding F as a function of H σ-2 into itself, it is not difficult to show that it a Lipschitz continuous function. Owing to Lemma 7.3 in Appendix, it is also a Lipschitz continuous function in H s -norm for all 0 ≤ s ≤ σ -2, provided u and v lie in B σ-2 R . We have indeed

F (u) -F (v) H s = f (|u| 2 )u -f (|u| 2 )v + f (|u| 2 )v -f (|v| 2 )v H s = f (|u| 2 )(u -v) H s + (f (|u| 2 ) -f (|v| 2 ))v H s ≤ κ f (|u| 2 ) H σ-2 u -v H s + κ v H σ-2 f (|u| 2 ) -f (|v| 2 ) H s ≤ κ Aχ f (c 2 u 2 H σ-2 ) u 2 H σ-2 u -v H s + α v H σ-2 α(f, AR 2 ) |u| 2 -|v| 2 H s ≤ κ Aχ f (c 2 R 2 )R 2 u -v H s + κ R α(f, AR 2 ) uū -uv + uv -vv H s ≤ κ Aχ f (c 2 R 2 )R 2 u -v H s + 2 κ 2 R 2 α(f, AR 2 ) u -v H s ≤ L u -v H s with 3 L = κ R 2 (Aχ f (c 2 R 2 ) + 2 κ α(f, AR 2 )).
The map e ih∆ being an isometry of H s for all s, it is clear that F τ is also Lipschitz continuous with the same constant L.

Derivatives with respect to u. Let s = σ or s = σ -2. Function F τ is C ∞ w.r.t. u from H s into itself, and we have for instance

F ′ τ (u)(v) = e -iτ ∆ F ′ (e iτ ∆ u)(e iτ ∆ v)
where 6) for f and f ′ and the identity e iτ ∆ u H s = u H s , we get the estimates for all τ ∈ [0, 1], all u ∈ B s R and all v ∈ H s F τ (u)

F ′ (u)(v) = -if (|u| 2 )v -if ′ (|u| 2 )(ūv + vu)u. From (
H s ≤ A 2 χ f (c 2 u 2 H s ) u 3 H s ≤ M 0 and F ′ τ (u)(v) H s ≤ A 3 f ′ (|e iτ ∆ u| 2 ) H s u 2 H s v H s + A f (|e iτ ∆ u| 2 ) H s v H s ≤ M 1 v H s
for some constants M 0 and M 1 . The second derivative of F takes the form

F ′′ (u)(v, w) = -if ′′ (|u| 2 )u(ūv + vu)(ūw + wu) -2if ′ (|u| 2 )(ūvw + vuw + wuv),
so that by similar arguments, we obtain

∀u ∈ B s R , ∀(v, w) ∈ H s × H s , F ′′ τ (u)(v, w) H s ≤ M 2 v H s w H s .
Note that the constants M 0 , M 1 and M 2 depend on f and R. These estimates can be easily generalized to higher derivatives F (l) τ , l = 3, . . . , ⌊σ/2⌋ for some constants M l .

Derivatives with respect to τ . The first derivative of F τ w.r.t. τ can be computed as follows:

dF τ (u) dτ = -ie -iτ ∆ ∆F (e iτ ∆ u) + e -iτ ∆ F ′ (e iτ ∆ u)(ie iτ ∆ ∆u) = -i∆F τ (u) + F ′ τ (u)(i∆u).
Note that here,

iF ′ τ (u)(∆u) -F ′ τ (u)(i∆u) = 0, so that the function u → dFτ (u) dτ is a function from H σ to H σ-2 . The H σ-2 -norm of F τ (u) for u ∈ B σ R can be estimated as follows dF τ (u) dτ H σ-2 ≤ ∆F τ (u) H σ-2 + F ′ τ (u)(i∆u) H σ-2 ≤ M 0 + M 1 R.
More generally, the j-th derivative w.r.t. τ reads 4

d j F τ (u) dτ j = j l=0 j l (-i) j-l ∆ j-l F (l) τ (u)(i∆u) l
and is a function from H σ into H σ-2j , provided σ ≥ 2j. Its H σ-2j -norm can be bounded as follows

d j F τ (u) dτ j H σ-2j ≤ j l=0 j l F (l) τ (u)(i∆u) l H σ-2l ≤ F τ (u) H σ + j l=1 j l F (l) τ (u)(i∆u) l H σ-2 ≤ (1 + R) j max l=0,...,j
M l . 4 We sometimes use the notation F (l) (u)(w j ) as a shorthand for F (l) (u) (w, . . . , w) j times .

The bounds of local truncation errors will finally require to estimate

d dτ F ′ τ (u)(v) = -i∆F ′ τ (u)(v) + F ′ τ (u)(i∆v) + F ′′ τ (u)(v, i∆u).
For u and v in B σ R , the following bound stems directly from previous inequalities

dF ′ τ (u) dτ H σ-2 ≤ 2M 1 R + M 2 R 2 .
H σ -H s inequality for derivatives of F τ . Proceeding as for F τ , it is straightforward to show that derivatives of F τ w.r.t. to u satisfy a Lipschitz estimate on B σ R : there exists a constant L > 0 such that, for all 0 ≤ 2j ≤ σ,

∀w ∈ B σ-2 R , ∀(u, v) ∈ B σ R × B σ R , F (j) (u)(w j ) -F (j) (v)(w j ) H σ-2 ≤ L u -v H σ-2 .
It then follows that

∀(u, v) ∈ B σ R × B σ R , d j F τ (u) dτ j - d j F τ (v) dτ j H σ-2j ≤ L 2 u -v H σ
for some constant L 2 . We finally collect in next proposition the findings of this Section.

Proposition 2.1. Let m = ⌊σ/2⌋. For s ∈ {σ, σ -2}, the function (τ, u) → F τ (u) = e -iτ ∆ F (e iτ ∆ u) is a well-defined function from [0, 1] × H s into H s , is C ∞ w.r.t. u
, and for all 0 ≤ j ≤ m, it is j-times differentiable with values in H σ-2j . Moreover, there exists a positive constant M such that for all τ ∈ [0, 1], all u ∈ B s R and all (v 1 , v 2 ) ∈ (H s ) 2 one has

F τ (u) H s ≤ M, F ′ τ (u)(v 1 ) H s ≤ M v 1 H s , F ′′ τ (u)(v 1 , v 2 ) H s ≤ M v 1 H s v 2 H s , (7)
and for all τ ∈ [0, 1] and all (u, v) ∈ (B σ R ) 2 , one has

d j F τ (u) dτ j H σ-2j ≤ M and dF ′ τ (u) dτ H σ-2 ≤ M. (8)
Finally, there exist positive constants L and L 2 such that for all τ ∈ [0, 1]

∀0 ≤ s ≤ σ -2, ∀(u, v) ∈ B σ-2 R × B σ-2 R , F τ (u) -F τ (v) H s ≤ L u -v H s , (9)
and for all 0

≤ j ≤ m, all τ ∈ [0, 1] ∀(u, v) ∈ B σ R × B σ R , d j F τ (u) dτ j - d j F τ (v) dτ j H σ-2j ≤ L 2 u -v H σ . ( 10
)
The constants M , L and L 2 depend on f , R and σ.

Stability estimates

In order to estimate local errors in a neighborhood of the exact solution, we need to ensure that the numerical solutions lie in B σ-2 R . Besides, stability estimates in various norms are required in the study of the error accumulation. These points are addressed in next lemma.

≤ s ≤ σ -2 Φ h (v) -Φ h (w) H s ≤ e εLh v -w H s and A h 1 (v) -A h 1 (w) H s ≤ εhe εLh v -w H s . Proof. Denoting v(t) = ϕ t V (v) and w(t) = ϕ t V (w), we have ẇ(t) = εF (w(t)) and v(t) = εF (v(t)) so that v(t) -w(t) H s ≤ v -w H s + ε t 0 F (v(τ )) -F (w(τ )) H s dτ.
As long as w(t) and v(t) remain in B σ-2 R , we have, according to ( 9)

F (v(τ )) -F (w(τ )) H s ≤ L v(τ ) -w(τ ) H s so that, by Gronwall lemma v(t) -w(t) H s ≤ e εLt v -w H s .
In particular, taking w = 0, we have v(h) H σ-2 ≤ 3R 4 e εLh ≤ R. Now, we immediately get

ϕ h V (v) -v -ϕ h V (w) + w H s ≤ ε L h e εLh v -w H s ,
and both statements then follow from the fact that e i h 2 ∆ is an isometry of H s .

Local truncation errors

Denoting t n = nh ≤ T /ε and u ε n = u ε (t n ), we define the local truncation errors as follows

δ n (ε, h) = Φ h (u ε n ) -u ε n+1 .
Comparing the Taylor expansions of u ε (t n + h) and Φ h (u ε n ) and using Proposition 2.1, we can now derive bounds for δ n (ε, h).

Expansion of the exact solution

The exact solution at time t + h can be written with the Duhamel formula as

u ε (t + h) = e ih∆ u ε (t) + εe ih∆ h 0 F τ (e -iτ ∆ u ε (t + τ ))dτ, or equivalently for v ε (t) = e -it∆ u ε (t) v ε (t + h) = v ε (t) + ε h 0 F t+τ (v ε (t + τ ))dτ. ( 11
)
We then use the following expansion of F τ (ũ + ṽ)

F τ (ũ + ṽ) = F τ (ũ) + F ′ τ (ũ)(ṽ) + 1 0 (1 -ξ)F ′′ τ (ũ + ξṽ)(ṽ 2 )dξ.
Inserting the expression of v ε (t + h) into the right-hand side of ( 11) and denoting v = v ε (t) for the sake of brevity, we obtain

v ε (t + h) = v + ε h 0 F t+τ v + ε τ 0 F t+τ 1 (v ε (t + τ 1 ))dτ 1 dτ = v + ε h 0 F t+τ (v)dτ + ε 2 h 0 τ 0 F ′ t+τ (v)F t+τ 1 (v ε (t + τ 1 ))dτ 1 dτ + ε 3 h 0 1 0 (1 -ξ)F ′′ t+τ (1 -ξ)v + ξv ε (t + τ ) dξ τ 0 F t+τ 1 (v ε (t + τ 1 ))dτ 1 2 dτ.
The second-order term in ε can be further expanded as

h 0 τ 0 F ′ t+τ (v)F t+τ 1 (v)dτ 1 dτ +ε h 0 τ 0 F ′ t+τ (v) 1 0 F ′ t+τ 1 (1 -ξ)v + ξv ε (t + τ 1 ) dξ τ 1 0 F t+τ 2 (v ε (t + τ 2 ))dτ 2 dτ 1 dτ.
Finally, we have with u = u ε (t)

u ε (t + h) = e ih∆ u + εe ih∆ h 0 F τ (u)dτ + ε 2 e ih∆ h 0 τ 0 F ′ τ (u)F τ 1 (u)dτ 1 dτ + ε 3 e ih∆ E 3 (u, ε, h), with E 3 (u, ε, h) = E 3,a (u, ε, h) + E 3,b (u, ε, h) where E 3,a (u, ε, h) = h 0 1 0 (1 -ξ)F ′′ τ (1 -ξ)u + ξe -iτ ∆ u ε (t + τ ) dξ τ 0 F τ 1 (e -iτ 1 ∆ u ε (t + τ 1 ))dτ 1 2 dτ, E 3,b (u, ε, h) = h 0 τ 0 F ′ τ (u) 1 0 F ′ τ 1 (1-ξ)u+ξe -iτ 1 ∆ u ε (t+τ 1 ) dξ τ 1 0 F τ 2 (e -iτ 2 ∆ u ε (t+τ 2 ))dτ 2 dτ 1 dτ. Since u ε (t) remains in B σ R/2 ⊂ B σ R ⊂ B σ-2 R
, we have by Proposition 2.1

E 3 (u ε (t), ε, h) H σ-2 ≤ 1 3 M 3 h 3 . ( 12
)

Expansion of numerical solutions

Assume here that u ∈ B σ-2 3R/4 and 0 < h ≤ h 0 where h 0 is defined in Lemma 3.1. Using that

d dh ϕ h V (u) = εF (ϕ h V (u))
we get immediately

ϕ h V (u) = u + εhF (u) + 1 2 (εh) 2 F ′ (u)F (u) + ε 3 h 0 (h -τ ) 2 2 F ′′ (ϕ τ V (u))(F (ϕ τ V (u))) 2 + F ′ (ϕ τ V (u))F ′ (ϕ τ V (u))F (ϕ τ V (u)) dτ,
which we may write as

ϕ h V (u) = u + εhF (u) + 1 2 (εh) 2 F ′ (u)F (u) + ε 3 E 3,V (u, ε, h) (13)
with

E 3,V (u, ε, h) = h 0 (h -τ ) 2 2 F ′′ (ϕ τ V (u))(F (ϕ τ V (u))) 2 + F ′ (ϕ τ V (u))F ′ (ϕ τ V (u))F (ϕ τ V (u)) dτ.
The Strang splitting solution can then be expanded as

Φ h (u) = e ih/2∆ e ih/2∆ u + εhF (e ih/2∆ u) + 1 2 (εh) 2 F ′ (e ih/2∆ u)F (e ih/2∆ u) + ε 3 E 3,V (e ih/2∆ u, ε, h) = e ih∆ u + εhF h/2 (u) + 1 2 (εh) 2 F ′ h/2 (u)F h/2 (u) + ε 3 e ih/2∆ E 3,V (e ih/2∆ u, ε, h). ( 14
)
Note that owing to Lemma 3.1, ϕ τ V (u) remains in B σ-2 R for 0 ≤ τ ≤ h so that, by using Proposition 2.1,

∀u ∈ B σ-2 3R/4 , ∀0 < h ≤ h 0 , E 3,V (u, ε, h) H σ-2 ≤ M 3 3 h 3 . ( 15
)

Bounds of local truncation errors

The local truncation error finally reads

δ n (ε, h) = εe ih∆ hF h/2 (u ε n ) - h 0 F τ (u ε n )dτ + ε 2 e ih∆ 1 2 h 2 F ′ h/2 (u ε n )F h/2 (u ε n ) - h 0 τ 0 F ′ τ (u ε n )F τ 1 (u ε n )dτ 1 dτ + ε 3 e ih/2∆ E 3,V (e ih/2∆ u ε n , ε, h) -e ih∆ E 3 (u ε n , ε, h) (16)
and we have to estimate each term individually.

First-order term in ε. We use the second-order Peano kernel κ 2 of the midpoint rule

hF h/2 (u) - h 0 F τ (u)dτ = h 3 1 0 κ 2 (τ ) d 2 dθ 2 F θ (u) θ=τ h
dτ where κ 2 is a scalar continuous function, so that, owing to (8) (note that σ ≥ 4)

hF h/2 (u) - h 0 F τ (u)dτ H σ-4 ≤ M 1 0 |κ 2 (τ )|dτ h 3 . ( 17
)
Second-order term in ε. We here use that

F τ 1 (u) = F h/2 (u) + τ 1 h/2 d dθ F θ (u)dθ and F ′ τ (u) = F ′ h/2 (u) + τ h/2 d dθ F ′ θ (u)dθ
and insert these expressions into the double integral term to get

h 0 τ 0 F ′ τ (u)F τ 1 (u)dτ 1 dτ = h 2 2 F ′ h/2 (u)F h/2 (u) + r 1
where

r 1 = h 0 τ 0 F ′ h/2 (u) τ 1 h/2 d dθ F θ (u)dθdτ 1 dτ + h 0 τ 0 τ h/2 d dθ F ′ θ (u)dθF h/2 (u)dτ 1 dτ + h 0 τ 0 τ h/2 d dθ F θ (u)dθ τ 1 h/2 d dθ F θ (u)dθdτ 1 dτ so that r 1 H σ-2 ≤ 1 4 M 2 h 3 + 1 32 M 2 h 4 ≤ 1 4 M 2 h 3 (1 + h 0 /8).
Third-order term in ε. Collecting previous estimates of this section and estimates (15) and ( 12), we obtain

∃C > 0, ∀0 < h ≤ h 0 , δ n (ε, h) H σ-4 ≤ Cεh 3 . ( 18
)
We end up this section by noticing that the following bound also holds (using the first-order Peano kernel)

∃ C > 0, ∀0 < h ≤ h 0 , δ n (ε, h) H σ-2 ≤ Cεh 2 , (19)
and for later use in our refined analysis of errors, we observe that

δ n (ε, h) = εe ih∆ Λ h (u ε n ) + ε 2 R h (u ε n ) (20) with Λ h (u ε n ) = hF h/2 (u ε n ) - h 0 F τ (u ε n )dτ, (21)
and

∀0 < h ≤ h 0 , Λ h (u ε n ) H σ-4 ≤ Ch 3 and R h (u ε n ) H σ-2 ≤ Ch 3 .

Global error estimates

The basic ingredient of the proofs of this section is the following telescopic identity

(Φ h ) n (u 0 ) -u ε n = n l=1 (Φ h ) n-l • Φ h (u ε l-1 ) -(Φ h ) n-l (u ε l ) .
We proceed in two steps: in the first one, we obtain ε-independent error estimates on the whole interval [0, T /ε] (in agreement with [START_REF] Chr | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF]) and in the second one we use these estimates in a more refined analysis, on one period and then on the whole interval.

H σ-4 -convergence

The following theorem is the formulation in our context of a result from [START_REF] Chr | On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations[END_REF].

Theorem 5.1. Let h 1 = min(h 0 , RL 4 C(e LT -1)
) where h 0 is defined in Lemma 3.1. The numerical solution given by the Strang splitting scheme for equation ( 2) with 0 ≤ h ≤ h 1 satisfies a second-order error bound in

H σ-4 ∃C > 0, (Φ h ) n (u 0 ) -u ε (t n ) H σ-4 ≤ C e LT -1 L h 2 for t n = nh ≤ T /ε. (22)
The constants L and C depend on σ, R and f , but are independent of ε.

Proof. As long as the numerical approximations (Φ h ) j (u ε (t k )) remain in B σ-2 3R/4 for 1 ≤ j +k ≤ n, stability estimates of Lemma 3.1 hold and the telescopic identity leads to

(Φ h ) n (u 0 ) -u ε (t n ) H σ-2 ≤ n l=1 e εLh(n-l) δ l-1 (ε, h) H σ-2
where δ l-1 (ε, h) H σ-2 can be bounded by ε Ch 2 (see ( 19)). Hence, since nh ≤ T /ε, we have straightforwardly

(Φ h ) n (u 0 ) -u ε (t n ) H σ-2 ≤ C e nεLh -1 L h ≤ C e LT -1 L h.
The boundedness in H σ-2 required by the stability lemma is ensured by induction by the previous error bound provided C e LT -1 L h ≤ R 4 . Now, Lemma 3.1 applies and gives stability in H σ-4 , so that owing to estimate (18), we get the second-order convergence in H σ-4 .

H σ-2m -convergence on one period

We now examine more closely the errors after one period. Let us first notice that, according to Theorem 5.1, (Φ h ) l ∈ B σ-2 3R/4 for all 0 ≤ l ≤ n with nh ≤ T /ε and hence (Φ h ) l ∈ B s 3R/4 for all 0 ≤ s ≤ σ -2.

Lemma 5.2. The following estimate holds

∀0 ≤ t ≤ 1, u ε (t) -e it∆ u 0 H σ ≤ εM. ( 23
)
Proof. The terms u ε (t) and e it∆ u 0 are the exact solutions of the differential equations

uε = i∆u ε + εF (u ε ), u ε (0) = u 0 and vε = i∆v ε , v ε (0) = u 0 .
By Duhamel formula and Proposition 2.1, we have immediately

u ε (t) -v ε (t) H σ = ε t 0 e i(t-τ )∆ F (u ε (τ ))dτ H σ ≤ εtM ≤ εM.
Lemma 5.3. Given u and v, assume that both sequences (Φ h ) l (u) l and (Φ h ) l (v) l lie in B σ-2 3R/4 for all 0 ≤ l ≤ n with nh ≤ 1 and 0 < h ≤ h 1 where h 1 is defined in Theorem 5.1. Then the map u → A h l (u) := (Φ h ) l (u) -e ilh∆ u satisfies a Lipschitz condition for all 0 ≤ s ≤ σ -2 of the form

A h l (u) -A h l (v) H s ≤ εLe εL u -v H s .
Proof. A telescopic identity gives

(Φ h ) l (u) -e ilh∆ u = l k=1 e ih(l-k)∆ Φ h -e ih∆ • (Φ h ) (k-1) (u) .
Hence,

A h l (u) -A h l (v) H s ≤ l k=1 A h 1 ((Φ h ) (k-1) (u) -A h 1 ((Φ h ) (k-1) (v))
H s and according to Lemma 3.1 , we have

A l h (u) -A l h (v) H s ≤ l k=1 εLhe Lεh (Φ h ) (k-1) (u)) -(Φ h ) (k-1) (v) H s ≤ l k=1 εLhe kLεh u -v H s ≤ εLe εL u -v H s .
Theorem 5.4. Let m = ⌊σ/2⌋. The numerical solution given by the Strang splitting scheme for equation ( 2) with step size 0 < h ≤ h 1 where h 1 is defined in Theorem 5.1, has a secondorder error bound in H σ-2m of the form

∃ Ĉ > 0, (Φ h ) n (u 0 ) -u ε (1) H σ-2m ≤ Ĉ(ε 2 h 2 + εh m ) for nh = 1. ( 24
)
The constant Ĉ depends on σ, R and f , but is independent of ε.

Proof. The proof proceeds in two steps.

Identification of the ε-error term. Replacing (Φ h ) (n-l) by e ih(n-l)∆ in the telescopic identity we get

(Φ h ) n (u 0 ) -u ε (1) = n l=1 e ih(n-l)∆ δ l-1 (ε, h) + r where r = n l=1 A h n-l Φ h (u ε l-1 ) -A h n-l (u ε l ) .
According to previous lemma and using (18), we can estimate r as follows:

r H σ-4 ≤ εLe εL n l=1 δ l-1 (ε, h) H σ-4 ≤ ε 2 Le εL Ch 2 .
In addition, according to (20), we have

n l=1 e ih(n-l)∆ δ l-1 (ε, h) = ε n l=1 e ih(n-l+1)∆ Λ h (u ε l-1 ) + r where r = ε 2 n l=1 e ih(n-l)∆ R h (u ε l-1 )
can again be bounded by Cε 2 h 2 in H σ-2 -norm. Finally, taking into account that (see Lemma 5.2) u ε l-1 -e i(l-1)h∆ u 0 H σ ≤ M ε and that Λ h is Lipschitz continuous with a Lipschitz constant of the form h 3 L2 (see (21-17) and Proposition 2.1) we have

(Φ h ) n (u 0 ) -u ε (1) -ε n l=1 e i(n-l+1)h∆ Λ h (e i(l-1)h∆ u 0 ) H σ-4 ≤ Const ε 2 h 2 .
Estimate of the ε-error term. From previous analysis, the main error is concentrated in the term n l=1 e i(n-l+1)h∆ Λ h (e i(l-1)h∆ u 0 ) which is of order εh 2 . For a finer estimation, we now proceed as follows:

n l=1 e i(n-l+1)h∆ Λ h (e i(l-1)h∆ u 0 ) = n-1 l=0 hF (lh+h/2) (u 0 ) - 1 0 F τ (u 0 )dτ = n-1 l=0 hF (lh+h/2) (u 0 ) - 1 0 F (τ +h/2) (u 0 )dτ = e -ih/2∆ E Rie (e ih/2∆ u 0 , h)
where E Rie denotes the error in the approximation by Riemann sums according to Lemma 7.1 and where we have right away taken into account that e inh∆ is the identity operator and that τ → F τ (u 0 ) is 1-periodic. From Lemma 7.1, we thus have

n l=1 e i(n-l+1)h∆ Λ h (e i(l-1)h∆ u 0 ) H σ-2m ≤ C Rie m d j dτ j F τ H σ-2m h m with C Rie m = 2 ζ(m) (2π) m .
Together with estimate (8) of Proposition 2.1, this completes the proof.

H σ-2m -convergence on the whole interval

It is now straightforward to get error estimates on the whole interval by considering Φ(u) = (Φ h ) n (u) as an integrator with step size 1, for which the "local error" is of size Ĉε(εh 2 + h m ) and which is obviously Lipschitz continuous. A telescopic identity with Φ iterated N = ⌊T /ε⌋ times then leads to an error of size ĈT (εh 2 + h m ). Finally, the solution at intermediate points (i.e. within an interval [n, n + 1] or [N, T /ε]) can then be obtained by composing Φ and Φ h .

Theorem 5.5. Let m = ⌊σ/2)⌋. The numerical solution given by the Strang splitting scheme for equation (2) with step size h > 0 such that 1/h ∈ N has a second-order error bound in H σ-2m of the form

∃ Ĉ > 0, (Φ h ) n (u 0 ) -u ε (1) H σ-2m ≤ ĈT (εh 2 + h m ) for nh ≤ T /ε. (25)
The constant Ĉ depends on σ, R and f , but is independent of ε.

Extension to general splitting methods

Assume now that

Φ h (u) = e ihα 1 ∆ • ϕ β 1 h V • • • • • e ihαr∆ • ϕ βrh V
, is a general splitting method of order p. More precisely, suppose that

δ n (ε, h) = Φ h (u ε n ) -u ε n+1
can be written as in (20)

δ n (ε, h) = εe ih∆ Λ h (u ε n ) + ε 2 R h (u ε n ) with Λ h (u ε n ) H σ-2p ≤ Ch p+1 and R h (u ε n ) H σ-2p+2 ≤ Ch p+1 .
Then, it is possible to identify through an ε-expansion the term Λ h . One has indeed

Λ h (u) = r j=1 β j he ih(γ j -1)∆ F e ih(1-γ j )∆ u - h 0 F τ (u)dτ,
where γ j = j k=1 α k and γ r = 1 by a first-order condition on the splitting method. By consistency again, s j=1 β j = 1, and one has

Λ h (u) = r j=1 β j hF (γ j -1)h (u) - h 0 F τ (u) so that n l=1 e i(n-l+1)h∆ Λ h (e i(l-1)h∆ u 0 ) = r j=1 β j n-1 l=0 hF (lh+γ j h) (u 0 ) - 1 0 F τ (u 0 )dτ = r j=1
β j e -iγ j h∆ E Rie (e iγ j h∆ u 0 , h)

Higher-order conditions on the splitting method then lead to the following expression

Λ h (u) = h p+1 1 0 κ p (τ ) d p F θ dθ p θ=τ h
(u)dτ so that Λ h is Lipschitz continuous with a Lipschitz constant of the L2 h p+1 . The proof derived for Strang splitting can then be readily adapted leading to the estimate

∃ Ĉ > 0, (Φ h ) n (u 0 ) -u ε (1) H σ-2m ≤ ĈT (εh p + h m ) for nh ≤ T /ε,
under the additional assumption that σ ≥ 2p.

Numerical experiments

In this section, we present numerical results of Strang and another fourth-order splitting methods when applied to NLS equation with initial conditions as follows:

i∂ t u ε = -∂ xx u ε + ε 2 cos(2x)|u ε | 2 u ε , 0 ≤ t ≤ T /ε, x ∈ T 2π (26)
u ε (0, x) = u 0 (x) = cos(x) + sin(x), x ∈ T 2π . (27)
Given that the wave function is enforced to be periodic in space for all positive times, it is in practice discretized by Fourier series as u ε (t, x) = Nx/2 -Nx/2+1 u ε k (t)e ikx , with N x = 256, such that errors originating from space discretization can be considered as negligible. The final time is taken here as T = π/4. The reference solution u ref used to assess numerical accuracy is obtained by a fourth-order time splitting method [START_REF] Yoshida | Construction of higher order symplectic integrators[END_REF] with a very small time-step ∆t = 2π/10 4 . The time-step for the Strang splitting is always taken under the form 2π/N , where N ∈ N * (note that in contrast with the assumption of the rest of the paper, the timeperiod is here 2π and not 1. This is obviously of no consequence, since a simple rescaling of time would lead to a period 1). Numerical errors are computed in discrete H s -norm of the wave function

error = u ref -u num H s :=   Nx/2 k=-Nx/2+1 (1 + |k| 2 ) s |( u ref ) k -( u num ) k | 2   1/2
, where u num is the numerical solution.

Strang splitting scheme

Figure 1 presents the errors as a function of step size h in L 2 -norm (left) and H 1 -norm (right) for ε = 2 -6 , 2 -7 , . . . , 2 -12 (from top to bottom). A line of slope 2 (black circle line) is drawn as reference. One can clearly observe that for fixed ε, the error scales like h 2 . On the other hand, for fixed h, the error scales like ε. The contribution to the error of the term εh 2 is perfectly apparent in Table 1.

Fourth order splitting scheme

Figure 2 now presents the errors as a function of step size h in L 2 norm (left) and H 1 norm (right) for ε = 2 -3 , 2 -4 , . . . , 2 -9 (from top to bottom) for the fourth-order splitting method of [START_REF] Yoshida | Construction of higher order symplectic integrators[END_REF]. Again, a line of slope 4 (black circle line) is given as a reference. Essentially the same conclusions as for Strang can be drawn, confirming the estimates of Section 6. The contribution to the error of the term εh 4 is made perfectly clear in Table 2.

Appendix.

An elementary result on Riemann sums for periodic functions

Lemma 7.1. For the function (τ, u) ∈ [0, 1] × H σ → F τ (u) ∈ H σ and given h > 0 and n ≥ 1 such that nh = 1, consider the error E Rie (u, h) := 1 n n-1 l=0 F lh (u) - 1 0 F τ (u)dτ (28)
in the approximation of the integral by its Riemann sum. Then the following statement hold for m = ⌊σ/2⌋

E Rie (u, h) H σ-2m ≤ 2 ζ(m) (2π) m sup τ ∈[0,1] d m dτ m F τ (u) H σ-2m h m . (29)

Two lemmas on products and functions in Sobolev spaces

Let us first recall the fractional Leibniz rule: Lemma 7.2. For all s > 0 and 1 < p < +∞, we have uv W s,p (T d ) ≤ α u L p 1 (T d ) v W s,q 1 (T d ) + α u W s,q 2 (T d ) v L p 2 (T d ) (30)

provided 1 p = 1 p 1 + 1 q 1 = 1 p 2 + 1 q 2
, q 1 , q 2 ∈ (1, +∞), p 1 , p 2 ∈ (1, +∞].

Proof. This inequality can be deduced from the standard fractional Leibniz rule stated in Sobolev spaces on R d . Now we are in a position to prove the following lemma, where all functional spaces are implicitly defined on the torus T d . Lemma 7.3. Let s ≥ 0 and σ > d 2 , with s ≤ σ. Then (i) for all u ∈ H s and v ∈ H σ , we have

uv H s ≤ κ u H s v H σ , (31)

Lemma 3 . 1 .

 31 Consider the map u → A h 1 (u) := Φ h (u) -e ih∆ u and let h 0 = log(4/3) Lε 0 . If v and w are two functions of B σ-2 3R/4 , then for all 0 ≤ h < h 0 , Φ h (v) and Φ h (w) are in B σ-2 R and the following statements hold for all 0

Figure 1 :

 1 Figure 1: Numerical errors for Strang splitting at time T /ε in L 2 -norm and H 1 -norm for ε = 2 -6 , 2 -7 , . . . , 2 -12 (from top to bottom). Slope 2 in indicated as the black circle line.

Figure 2 :

 2 Figure 2: Numerical errors for Yoshida splitting at time T /ε in L 2 -norm and H 1 -norm for ε = 2 -6 , 2 -7 , . . . , 2 -12 (from top to bottom). Slope 4 in indicated as the black circle line.

h = 2π/2 4 h 1 0E

 41 = 2π/2 5 h = 2π/2 6 h = 2π/2 7 h = 2π/2 8 h = 2π/2 9 h = 2π/2 10 ε = 2 -3 4.777E-02 2.463E-02 2.193E-03 2.452E-04 1.719E-05 1.141E-06 7.255E-08 ε = 2 -4 2.375E-02 1.231E-02 1.048E-03 1.108E-04 7.854E-06 5.112E-07 3.230E-08 ε = 2 -5 1.186E-02 6.153E-03 5.180E-04 5.390E-05 3.834E-06 2.481E-07 1.564E-08 ε = 2 -6 5.926E-03 3.076E-03 2.582E-04 2.676E-05 1.905E-06 1.231E-07 7.758E-09 ε = 2 -7 2.963E-03 1.538E-03 1.290E-04 1.336E-05 9.511E-07 6.143E-08 3.871E-09 ε = 2 -8 1.481E-03 7.691E-04 6.449E-05 6.676E-06 4.754E-07 3.070E-08 1.934E-09 ε = 2 -9 7.407E-04 3.846E-04 3.224E-05 3.338E-06 2.377E-07 1.535E-08 9.670E-10 ε = 2 -3 2.14E-01 1.19E-01 6.80E-03 1.02E-03 2.98E-05 2.13E-06 1.40E-07 ε = 2 -4 1.06E-01 5.92E-02 2.28E-03 2.83E-04 1.23E-05 8.23E-07 5.27E-08 ε = 2 -5 5.30E-02 2.96E-02 9.49E-04 9.92E-05 5.79E-06 3.77E-07 2.39E-08 ε = 2 -6 2.65E-02 1.48E-02 4.47E-04 4.29E-05 2.85E-06 1.84E-07 1.16E-08 ε = 2 -7 1.32E-02 7.38E-03 2.20E-04 2.05E-05 1.42E-06 9.15E-08 5.76E-09 ε = 2 -8 6.62E-03 3.69E-03 1.10E-04 1.02E-05 7.09E-07 4.57E-08 2.87E-09 ε = 2 -9 3.31E-03 1.85E-03 5.48E-05 5.06E-06 3.55E-07 2.28E-08 1.44E-09Proof. Let (τ, u) ∈ [0, 1] × H σ → k∈Z e i2πkτ Fk (u)be the Fourier expansion of F τ . By definition of Fourier coefficients, F τ (u)dθ = F0 (u). As for the numerical counterpart of this integral, Rie (u, h)H σ-2m ≤ q∈Z * Fnq (u) H σ-2m .Besides, owing to the regularity ofF τ ∀k ∈ Z * , Fk (u) H σ-2m ≤ sup τ ∈[0,1] d m dτ m F τ (u) H σ-2m (2π|k|) mand it follows thatE Rie (u, h) H σ-2m ≤ 2 sup τ ∈[0,1] d m dτ m F τ (u) H σ-2m q∈N * (2πnq) -m = 2 sup τ ∈[0,1] d m dτ m F τ (u) H σ-2m ζ(m) (2πn) m

Table 1 :

 1 Errors of Strang splitting for NLS on T 2π in L 2 norm (upper) and H 1 norm.

		h = 2π/2 4	h = 2π/2 5	h = 2π/2 6	h = 2π/2 7	h = 2π/2 8
	ε = 2 -6	9.037E-03	3.279E-03	6.121E-04	1.489E-04	3.700E-05
	ε = 2 -7	4.519E-03	1.639E-03	3.060E-04	7.444E-05	1.850E-05
	ε = 2 -8	2.259E-03	8.196E-04	1.530E-04	3.722E-05	9.249E-06
	ε = 2 -9	1.130E-03	4.098E-04	7.648E-05	1.861E-05	4.624E-06
	ε = 2 -10	5.648E-04	2.049E-04	3.824E-05	9.304E-06	2.312E-06
	ε = 2 -6	1.415E-02	9.691E-03	1.230E-03	2.930E-04	7.244E-05
	ε = 2 -7	7.077E-03	4.845E-03	6.142E-04	1.464E-04	3.621E-05
	ε = 2 -8	3.538E-03	2.423E-03	3.071E-04	7.319E-05	1.810E-05
	ε = 2 -9	1.769E-03	1.211E-03	1.535E-04	3.660E-05	9.052E-06
	ε = 2 -10	8.846E-04	6.056E-04	7.676E-05	1.830E-05	4.526E-06

Table 2 :

 2 Errors of Yoshida splitting for NLS on T 2π in L 2 norm (upper) and H 1 norm.

Wolfgang Pauli Institute, Nordbergstrasse 15, 1090 Vienna, Austria, E-mail:: sunny5zhang@gmail.com (Yong Zhang)[START_REF] Alinhac | Pseudo-differential operators and Nash-Moser theorem[END_REF] The assumption T d = [0, 2π] d can be relaxed, provided the corresponding spectrum of ∆ remains a subset of ωN for some ω > 0.

Such step sizes are said to be resonant and can lead to exponential error growth (see Weideman and Herbst[START_REF] Weideman | Split-step methods for the solution of the nonlinear Schrödinger equation[END_REF]). However, this possible very long-time instability does not contradict the convergence results given. Instabilities are indeed typically observed on intervals of length T /ε 2 in this scaling.

Constants κ and α(f, AR 2 ) are defined in Lemma 7.3 in Appendix.

(ii) for all u, v ∈ H σ with u H σ ≤ R, v H σ ≤ R and for all smooth function f from R to R, we have

Proof. Recall that, by σ > d 2 , we have the Sobolev embedding

Hence, in the case s = 0, the result is obvious since we have

Consider now the case s > 0 and let us prove (31). If s > d 2 , the result is well-known since H s is an algebra. If 0 < s < d 2 , we apply (30) with the admissible set of coefficients

and use the Sobolev embeddings H σ ֒→ L ∞ , H s ֒→ L p 1 and H σ ֒→ W s,q 1 . We have then

we obtain the same estimate by applying (30) with

for µ > 0 small enough such that we also have the embeddings H s ֒→ L p 1 and H σ ֒→ W s,q 1 . Next, to prove (32), we consider the Taylor formula

and use a tame estimate in H σ (see [START_REF] Chartier | Solving highly-oscillatory NLS with SAM: numerical efficiency and geometric properties[END_REF]). Hence, applying (i) yields

The proof of the lemma is complete.