Christophe Chalons
email: christophe.chalons@uvsq.fr

Frédéric Coquel
email: frederic.coquel@cmap.polytechnique.fr

A new comment on the computation of non conservative products using Roe-type path conservative schemes

may come

A new comment on the computation of non conservative products using Roe-type path conservative schemes

Christophe Chalons, Frédéric Coquel

Introduction

We are interested in hyperbolic equations of the form

∂ t u + A(u)∂ x u = 0, x ∈ R, t ∈ R +,⋆ , u(x, 0) = u 0 (x), (1)
where u(x, t) ∈ R p is the unknown and u 0 represents the initial data, and supplemented with the validity of an entropy inequality

∂ t U(u) + ∂ x F (u) ≤ 0, (2)
where (U, F ) is an entropy-entropy flux pair (meaning T ∇UA = T ∇F with U strictly convex). From a theoretical point of view, it is well known that the solutions can develop discontinuities in finite time even if u 0 is smooth. It is therefore necessary to understand (1)-( 2) in a weak sense in order to define a notion of discontinuous weak solution which makes sense mathematically and physically. If (1) is a system of conservation laws, that is if there exists a flux 1 function f : R p → R p such that A = ∇f , the notion of weak solution in the sense of distributions is used to define the solutions of ( 1)- [START_REF] Abgrall | Numerical approximation of a compressible multiphase system[END_REF]. Assuming in addition that the characteristic fields are linearly degenerate or genuinely non linear, existence and uniqueness results have been obtained for initial data with bounded variation and close enough to a constant state (see for instance [START_REF] Serre | Systèmes de Lois de Conservation I et II[END_REF]).

From a numerical point of view, there exists in this situation a large number of finite volume methods to approximate the smooth and discontinuous solutions of (1)- [START_REF] Abgrall | Numerical approximation of a compressible multiphase system[END_REF]. The most famous of them are certainly the Godunov and Godunov-type methods whose fundamental ingredient is the (exact or approximate) Riemann solution, namely the solution with an initial data made of two constant states separated by a discontinuity (see for instance [START_REF] Godlewsky | Numerical approximation of hyperbolic systems of conservation laws[END_REF], [START_REF] Toro | Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction[END_REF]). The situation is much more intricate when the matrix A is not the Jacobian matrix of a flux function since the theory of distributions does not apply. The non conservative product A(u)∂ x u has indeed a loose sense at a point of discontinuity of the solution u.

To overcome this difficulty, Dal Maso, LeFloch and Murat [START_REF] Maso | Definition and weak stability of a non conservative product[END_REF] introduced for functions with bounded total variation a notion of weak solution that allows to define the non conservative product A(u)∂ x u and that generalizes the notion of weak solution in the sense of distributions for conservative systems. It is based on the theoretical notion of "path". Importantly, a path acts as a new degree of freedom of the model and a first difficulty consists in choosing the family of paths. Similarly to the conservative setting, this definition leads to existence and uniqueness results of weak solutions to (1)-( 2) under certain assumptions (see for instance [START_REF] Lefloch | Existence theory for nonlinear hyperbolic systems in nonconservative form[END_REF]). At the numerical level, imposing the path in the numerical method is still an open and very ambitious problem because of the sensitiveness of the traditional methods with respect to the choice of the path and the usual discretisation parameters, see for instance [START_REF] Hou | Why nonconservative schemes converge to wrong solutions: error analysis[END_REF] and the more recent review [START_REF] Lefloch | Numerical methods with controlled dissipation for small-scale dependent shocks[END_REF]. Several attempts to impose the so-called "path" within the numerical scheme have been tried. Without any attempt to be exhaustive, they proved to be very successfull in [START_REF] Berthon | Contribution à l'analyse numérique des équations de Navier-Stokes compressibles à deux entropies spécifiques[END_REF][START_REF] Berthon | Schéma nonlinéaire pour l'aproximation numérique d'un système hyperbolique non conservatif[END_REF][START_REF] Berthon | Nonlinear projection methods for multientropies Navier-Stokes systems[END_REF], [START_REF] Chalons | Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes[END_REF][START_REF] Chalons | Euler equations with several independent pressure laws and entropy satisfying explicit projection schemes[END_REF] (see also the references therein) but the strategy unfortunately highly relies on the structure of the models studied in the papers and generalizing the approach is not clear at all. More recently, C. Parés and collaborators developped in [START_REF] Parés | Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework[END_REF] and later works the so-called "path-conservative" schemes. Since then, it has been proved that this approach generally fails in converging to the right solutions, see [START_REF] Abgrall | A comment on the computation of non conservative products[END_REF], [START_REF] Castro | Why many theories of shock waves are necessary: Convergence error in formally pathconsistent schemes[END_REF]. Actually, such a failure happens for any numerical method in which the small scale effects are not controlled, either if it is path-conservative or not. At last, a different strategy from the one proposed in the present contribution and that significantly reduces these convergence errors for the same problem has been introduced in [START_REF] Fjordholm | Accurate numerical discretizations of nonconservative hyperbolic systems[END_REF] and generalized to general nonconservative systems within the framework of path-conservative methods in [START_REF] Castro | Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems[END_REF].

In [START_REF] Abgrall | A comment on the computation of non conservative products[END_REF], the authors more precisely consider the usual gas dynamics equations in Lagrangian coordinates and show that path-conservative schemes are not convergent when applied to a non conservative version of these equations. In the present paper, which structure follows closely Abgrall and Karni's [START_REF] Abgrall | A comment on the computation of non conservative products[END_REF], we consider the same equations and show how to slightly modifiy the usual "pathconservative" schemes to compute correctly the solutions of the non conservative formulation of the gas dynamics equations. It is based on a slight modification of the underlying averaging procedure by introducing modified cells as already introduced in several quite recent contributions, see for instance [START_REF] Zhong | Computational methods for propagating phase boundaries[END_REF], [START_REF] Chalons | Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling[END_REF], [START_REF] Chalons | Capturing infinitely sharp discrete shock profiles with the Godunov scheme[END_REF], [START_REF] Chalons | Computing material fronts with a Lagrange-Projection approach[END_REF], [START_REF] Bachmann | Random Sampling Remap For Compressible Two-Phase Flows[END_REF], for a non exhaustive list. We also refer the reader to [START_REF] Abgrall | Numerical approximation of a compressible multiphase system[END_REF] for a hybrid Roe-Glimm scheme which spirit is similar in some sense to the strategy we derive here.

The content of the first four sections is well-known but necessary to set up the problem and the notations for the sake of clarity. The main contribution is thus contained in Sections 5 and 6 where the method is described and numerical experiments are proposed. At last, Section 7 contains some concluding remarks.

The model under consideration

We consider the following gas dynamics equations in Lagrangian coordinates :

   ∂ t τ -∂ x u = 0, ∂ t u + ∂ x p = 0, ∂ t E + ∂ x pu = 0, ( 3
)
where t is the time, x the spatial coordinate, τ > 0 represents the inverse of a density, u is the velocity and p = p(τ, e) > 0 is the pressure of the fluid. Here e > 0 denotes the internal specific energy and satisfies E = e + u 2 /2. In order to illustrate our purpose, it will be sufficient to focus on the case of a perfect gas equation of state. It will much simplify the forthcoming calculations but more general equations of state could be considered as well. Therefore we set p(τ, e) = (γ -1)e/τ where γ > 1 is the so-called adiabatic coefficient. From a mathematical point of view, it is well known that (3) is strictly hyperbolic with eigenvalues λ 0 = 0 and λ ± = ±c, c = γp/τ . Moreover, the characteristic field associated with λ 0 is linearly degenerate and the characteristic fields associated with λ ± are genuinely nonlinear [START_REF] Godlewsky | Numerical approximation of hyperbolic systems of conservation laws[END_REF]. For the sake of conciseness, we introduce the following short form

∂ t u(x, t) + ∂ x f (u(x, t)) = 0, (4)
where the flux function f finds a natural definition with respect to the conservative unknowns u = (τ, u, E). At last, we supplement (4) with an initial data u 0 at time t = 0, that is to say u(x, t = 0) = u 0 (x).

A particular initial data is given by a Riemann initial data connecting two constant states u L and u R in the phase space (τ and p are positive) as follows

u 0 (x) = u L if x < 0, u R if x > 0. ( 5
)
Non conservative formulation. Easy calculations show that for smooth solutions system (3) can be equivalently recast under the following non conservation form

   ∂ t τ -∂ x u = 0, ∂ t u + ∂ x p = 0, ∂ t e + p∂ x u = 0, (6)
where only the last equation has been changed. Using a chain rule argument, we also introduce a short form for (6), namely

∂ t v(x, t) + A(v)(x, t)∂ x v(x, t) = 0, (7)
where v = (τ, u, e) and

A(v) =   0 -1 0 ∂ τ p(τ, e) 0 ∂ e p(τ, e) 0 p(τ, e) 0   .
Note that the mapping u → v is one-to-one so that the proposed change of variable is admissible. In particular, one can naturally define v 0 at time t = 0 from u 0 , and thus v(x, t = 0) = v 0 (x).

Jump relations and admissible shock solutions

It is well-known that the solutions of hyperbolic systems generally develop discontinuities in finite time. Considering system (4), a discontinuity

u(x, t) = u -if x < σt u + if x > σt
between two states u -and u + and propagating with velocity σ is a weak solution if and only if it satisfies the Rankine-Hugoniot jump relations

-σ(u + -u -) + f (u + ) -f (u -) = 0.
After some calculations, these relations write equivalently

   σ(τ + -τ -) + (u + -u -) = 0, -σ(u + -u -) + (p + -p -) = 0, -σ(e + -e -) + 1 2 (p + + p -)(u + -u -) = 0, (8)
while usual admissibility criteria based on entropy consideration or related Lax criterion impose the following compatibility condition σ(τ + -τ -) > 0, for shock discontinuities (σ = 0). We refer for instance the reader to [START_REF] Godlewsky | Numerical approximation of hyperbolic systems of conservation laws[END_REF] for these more than classical considerations. Note however that the former criterion will be a key ingredient in the definition of our numerical strategy.

Non conservative formulation.

If one now considers the non conservative formulation [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF], the theory of distributions to define weak solutions does not make sense. To overcome this difficulty, Dal Maso, LeFloch and Murat [START_REF] Maso | Definition and weak stability of a non conservative product[END_REF] introduced a generalization of the notion of weak solution in the sense of distributions that allows to define the non conservative products. More precisely, it is based on the theoretical notion of path. A path φ :

[0, 1] × R 3 × R 3 → R 3 associated with (7) satisfies the properties φ(0, v -, v + ) = v -and φ(1, v -, v + ) = v + and
serves to define the generalized Rankine-Hugoniot relations associated with a discontinuity

v(x, t) = v -if x < σt v + if x > σt
between two states v -and v + and propagating with velocity σ by setting

-σ(v + -v -) + 1 0 A(φ(s, v -, v + ))∂ s φ(s, v -, v + )ds = 0. ( 9
)
As expected, this definition does not depend on the path when the matrix A is the Jacobian matrix of a flux function like in formulation ( 4) and leads to the previous and usual Rankine-Hugoniot relations in such a situation. A very simple choice of path which will be considered in the following is defined for all v -and v + such that σ(τ + -τ -) > 0 and is linear with respect to τ , u and p, namely

   τ (s) = τ -+ s(τ + -τ -), u(s) = u -+ s(u + -u -), p(s) = p -+ s(p + -p -),
for all s ∈ [0, 1]. Easy calculations then show that the generalized jump relations (9) boil down to (8) so that both conservative and non conservative formulas are equivalent for such a choice of path, see [START_REF] Abgrall | A comment on the computation of non conservative products[END_REF].

Original Roe-type path conservative schemes

We briefly recall in this section the Roe-type path conservative schemes proposed in [START_REF] Parés | Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework[END_REF] to approximate the solutions of non conservative systems. It was shown in [START_REF] Abgrall | A comment on the computation of non conservative products[END_REF] that such a strategy fails in producing good approximations of admissible shock solutions when applied to system [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF]. The next section shows how to slightly modify this approach to make the numerical solutions perfectly match the exact ones. Let us first introduce some notations. We denote by ∆x and ∆t the space and time steps, respectively. We define the mesh interfaces x j+1/2 = j∆x for j ∈ Z and the intermediate times t n = n∆t for n ∈ N. We also define the mid-points x j = (x j-1/2 + x j+1/2 )/2 for j ∈ Z.

At each time t n , v n j represents an approximation of the exact solution on the interval [x j-1/2 , x j+1/2 ), j ∈ Z, and we look for an approximation of the solution at time t n+1 . At time t = 0 we set

v 0 j = 1 ∆x x j+1/2
x j-1/2 v 0 (x)dx, for all j ∈ Z.

According to [START_REF] Toumi | A weak formulation of Roe's approximate Riemann solver[END_REF] and [START_REF] Parés | Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework[END_REF], Roe-type path conservative schemes associated with [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF] and a given path φ are based on a Roe linearization A φ satisfying the properties 1. for all v L and v R ,

A φ (v L , v R ) has 3 distinct eigenvalues, 2. for all v, A φ (v, v) = A(v),
3. for all v L and v R ,

A φ (v L , v R )(v R -v L ) = 1 0 A(φ(s, v L , v R ))∂ s φ(s, v L , v R )ds.
In the sequel, we will follow [START_REF] Munz | On Godunov-Type Schemes for Lagrangian Gas Dynamics[END_REF] and set

A φ (v L , v R ) = A(v), v = v(v L , v R ) = (τ , u, e) with τ = τ L + τ R 2 , u = u L + u R 2 , e = p τ γ -1 and p = p L + p R 2 ,
which is easily seen to satisfy the three properties. The approximate Riemann solution constructed from the Roe linearization, namely the exact solution to the linear Riemann problem

   ∂ t v(x, t) + A φ (v L , v R )∂ x v(x, t) = 0, v(x, t = 0) = v L if x < 0, v R if x > 0, is then given by v(x/t; v L , v R ) =        v L if x/t < -σ(v L , v R ), v * L if -σ(v L , v R ) < x/t < 0, v * R if 0 < x/t < σ(v L , v R ), v R if x/t > σ(v L , v R ), (10)
where the left and right intermediate states are easily obtained from the left and right eigenvectors l k and

r k , k = 1, 2, 3 of A φ (v L , v R ), respectively, namely v * L = (v R , l 1 )r 1 + 3 k=2 (v L , l k )r k , v * R = 2 k=1 (v R , l k )r k + (v L , l 3 )r 3 , and σ(v L , v R ) = c(v(v L , v R )) = γp/τ (v L , v R ).
For the sake of clarity, it will be useful to have in mind the wave pattern of this solution, which is recalled on the next figure.

-

σ(v L , v R ) σ(v L , v R ) 0 v * L v * R v L v R x = 0
Once the solution is defined, we denote by x → ṽ(x, t) the piecewise constant approximate solution obtained by glueing together the Roe-type approximate solutions at each interface, that is to say ṽ

(x, t) = v((x -x j+1/2 )/t; v n j , v n j+1 ) for all (x, t) ∈ [x j , x j+1 ) × [0, ∆t), j ∈ Z, n ∈ N.
The Roe-type path conservative scheme is then defined according to [START_REF] Parés | Numerical Methods for Nonconservative Hyperbolic Systems: A Theoretical Framework[END_REF] as any Godunov-type scheme by averaging the solution on each cell [x j-1/2 , x j+1/2 ), namely

v n+1 j = 1 ∆x x j+1/2
x j-1/2 ṽ(x, ∆t)dx, [START_REF] Castro | Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems[END_REF] under the CFL restriction ∆t max

j∈Z |σ(v n j , v n j+1 )| ≤ ∆x 2 . ( 12
)
Easy calculations give the following form for the update formula,

v n+1 j = v n j - ∆t ∆x D + j-1/2 + D - j+1/2 , D ± j+1/2 = A ± φ (v n j , v n j+1 )(v n j+1 -v n j ),
where ± refers to the positive and negative part of the matrices. In the sequel, we will also use the notation σ n j+1/2 = σ(v n j , v n j+1 ).

Modified Roe-type path conservative schemes

In this section, we present a slight modification of the original Roe-type path conservative scheme in order to compute the admissible shocks properly. By slight, it is meant that only the average step of the method will be changed.

The proposed strategy was already introduced in several quite recent contributions, see for instance [START_REF] Zhong | Computational methods for propagating phase boundaries[END_REF], [START_REF] Chalons | Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling[END_REF], [START_REF] Chalons | Capturing infinitely sharp discrete shock profiles with the Godunov scheme[END_REF], [START_REF] Chalons | Computing material fronts with a Lagrange-Projection approach[END_REF], [START_REF] Bachmann | Random Sampling Remap For Compressible Two-Phase Flows[END_REF], for a non exhaustive list. It involves modified cells and random sampling, which objective is to get sharp discontinuities, i.e. without numerical diffusion. Controlling the numerical diffusion associated with the average step of Roe-type path conservative schemes, and more generally of any Godunov-type scheme, is a key point to approximate shocks in non conservative systems. We refer for instance the reader to [START_REF] Berthon | Contribution à l'analyse numérique des équations de Navier-Stokes compressibles à deux entropies spécifiques[END_REF], [START_REF] Berthon | Schéma nonlinéaire pour l'aproximation numérique d'un système hyperbolique non conservatif[END_REF], [START_REF] Berthon | Nonlinear projection methods for multientropies Navier-Stokes systems[END_REF], [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF], [START_REF] Chalons | Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes[END_REF], [START_REF] Chalons | Euler equations with several independent pressure laws and entropy satisfying explicit projection schemes[END_REF] for more details. As already underlined in the previous section, the Roe-type path conservative scheme defines the update value at the new time step by simply averaging on each fixed cells (x j-1/2 , x j+1/2 ) and at time t = ∆t the solution made of the juxtaposition of the Roe-type approximate Riemann solutions defined by ( 9), according to [START_REF] Castro | Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems[END_REF]. Instead, we propose to average the solution at time t = ∆t on modified cells (x j-1/2 , xj+1/2 ) defined by xj+1/2 = x j+1/2 + s j+1/2 ∆t, where s j+1/2 has to be defined. Note that s j+1/2 is expected to represent an approximation of the speed of propagation of an admissible shock in the Riemann problem set at interface x j+1/2 , so that we will have s n j+1/2 = -σ n j+1/2 or s n j+1/2 = +σ n j+1/2 . More precisely, we propose to set

v n+1 j = 1 ∆x j x j+1/2
x j-1/2 ṽ(x, ∆t)dx, [START_REF] Chalons | Euler equations with several independent pressure laws and entropy satisfying explicit projection schemes[END_REF] with ∆x j = xj+1/2 -xj-1/2 , as illustrated on the figure below.

t n t n+1 xj-1/2 xj+1/2 xj+3/2 v n+1 j v n+1 j+1 x j-3/2 x j-1/2 x j+1/2 x j+3/2
x j+5/2 In order to recover the initial mesh and define v n+1 j without introducing numerical diffusion, we then proceed with a random sampling procedure similar to the one proposed in the random choice method [START_REF] Collela | Glimm's method for gas dynamics[END_REF]. More precisely, let be given (a n ) n a well-distributed random sequence within (0, 1) (e.g. the van der Corput sequence), we simply set

v n+1 j =      v n+1 j-1 if a n+1 ∈ (0, ∆t ∆x s n,+ j-1/2 ) v n+1 j if a n+1 ∈ [ ∆t ∆x s n,+ j-1/2 , 1 + ∆t ∆x s n,- j+1/2 ) v n+1 j+1 if a n+1 ∈ [1 + ∆t ∆x s n,- j+1/2 , 1) (14)
where we have set s n,+ j+1/2 = max(s n j+1/2 , 0) and s n,- j+1/2 = min(s n j+1/2 , 0).

Definition of s n j+1/2 . In agreement with the selection principle σ(τ + -τ -) > 0 introduced in Section 3, we simply set

s n j+1/2 = -σ n j+1/2 if (τ n j+1 -τ n j ) ≤ 0, σ n j+1/2 if (τ n j+1 -τ n j ) > 0.
On the other hand, if the Riemann solution at interface x j+1/2 consists in a stationary contact discontinuity (u j = u j+1 and p j = p j+1 ), which we write here

|u j -u j+1 | ≤ ǫ and |p j -p j+1 | ≤ ǫ (15)
for a given threshold ǫ, we naturally set s n j+1/2 = 0. In practice, we set ǫ = 1e-8.

We thus have the following result.

Theorem 5.1 Under the CFL restriction [START_REF] Chalons | Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes[END_REF], the scheme defined by ( 13)-( 14) is consistent with [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF] in the following sense :

(i) Constant state : Assume that v := v n j-1 = v n j = v n j+1 , then v n+1 j = v.
(ii) Isolated shock discontinuity : Let v -and v + be two constant states that can be connected by an admissible discontinuity in the sense of Section 3. We denote by σ the speed of propagation. Assume that v 0 j = v -if j ≤ 0 and v 0 j = v + if j > 0. Then the scheme defined by ( 13)-( 14) is equivalent to Glimm's random choice scheme and then converges to the exact solution of [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF] given by v(x, t) = v -if x < σt and v(x, t) = v + if x > σt. In particular, we have v n j ∈ {v -, v + } ∀ j ∈ Z and ∀ n ∈ N so that no spurious state is introduced in the numerical solution.

Proof. The proof of the first point is trivial. Regarding the second point, it suffices to recall that the Roe linearization leads to an exact solution when it corresponds to an isolated shock wave. The rest of the proof is immediate and the convergence result of Glimm's scheme can be found for instance in [START_REF] Serre | Systèmes de Lois de Conservation I et II[END_REF].

The second point of the theorem shows in particular that the proposed scheme is able to compute properly and without numerical diffusion isolated shock waves associated with the non conservative system [START_REF] Berthon | Why many theories of shock waves are necessary : Kinetic relations for nonconservative systems[END_REF]. Actually, it coincides with the random choice method in such a specific case. The next section devoted to numerical experiments shows that the numerical solutions perfectly match the exact ones also for more general initial data still guaranteeing no numerical diffusion.

Numerical experiments in Lagrangian and Eulerian coordinates

Several test cases are proposed in this section to illustrate the behavior of the proposed numerical scheme. The adiabatic coefficient is set to be equal to γ = 1.4. We compare the solutions given by two numerical schemes, namely the original path-conservative scheme and the proposed numerical scheme. We use a mesh with 300 cells per unit of length and a CFL restriction of 0.45. The results are displayed with the x coordinate on the horizontal axis, and the unknowns specified in the figure captions on the vertical axis. The results are first given in Lagrangian coordinates, see Figures 1,2, 3 and 4, but also in Eulerian coordinates, see Figures 7, and8, following the direct extension of the proposed method described in appendix A. Note that the first two test cases below have been cooked up in such a way that exact solutions are either an isolated discontinuity or two shock discontinuities starting from the same right state, which explains the very specific values. In addition, these values can be understood in dimensionless form. The last test case in Eulerian coordinates is more realistic, it is inspired from the first test case of [START_REF] Abgrall | Numerical approximation of a compressible multiphase system[END_REF] and involves a very large pressure jump across the shock.

Test 1. The first test case consists in an isolated shock discontinuity :

(τ, u, p) 0 (x) = (2.09836065573770281, 2.3046638387921279, 1.0) if x < 0, (8.0, 0.0, 0.1) if x > 0.
The speed of propagation equals 0.3905124837953326544238. The final time of simulation is t = 0.5. As expected, we observe on Figure 1 that the original path-conservative scheme fails in producing good numerical results and creates spurious additional waves. On the contrary, Figures 1 and2 show that the proposed numerical scheme captures the discontinuity without numerical diffusion since again, it coincides with the random choice method in this case. It is thus convergent and first-order accurate as proved in [START_REF] Serre | Systèmes de Lois de Conservation I et II[END_REF]. (τ, u, p) 0 (x) = (5.0, 3.323013993227, 0.481481481481) if x < 0, (8.0, 0.0, 0.1) if x > 0.

The 3-shock is the same as in the previous test case. The density of the 1shock jumps from 5.0 to 3.0 and the discontinuity propagates with velocity -0.509175077217. The final time of simulation is t = 0.5. Again, we observe on Figure 3 that the original path-conservative scheme fails in producing good numerical results (the constant states are not properly restored), while the proposed numerical scheme provides the right states and captures the shock discontinuities without numerical diffusion. We also observe on Figure 3 that the position of the 2-shock is shifted with respect to the real position, which comes from the proposed random sampling. Actually, the position gets closer and closer to the right position when the mesh is refined as observed on Figure 4 for a 800-point mesh. Regarding the intermediate constant states in the solutions, Figure 5 proposes a zoom of the plateau by giving the errors between the exact and approximate values at given space positions. It shows that both values are extremely close for the three components τ , u and p. At last, Figure 6 gives the L 1 -errors for different meshes and for the three variables τ , u and p. The rate of convergence appears to be less than one and we postulate that it comes from the numerical diffusion introduced across the contact discontinuity. We follow this section by giving the results of the same two test cases but in Eulerian coordinates, see appendix A for more details. We plot ρ, u and p and the final times are t = 0.1 for both test cases. Note that the final time is shorter than in Lagrangian coordinates in order for the waves to remain in the computational domain. The meshes are made of 300 points and the solutions given by our scheme are now compared to the ones given by the usual method of Roe applied to the conservative equations. The latter allows for a proper computation of the left and right states of the shocks with the usual numerical diffusion in between. The objective is indeed to compare the numerical diffusion of both strategies. As it can be clearly seen, these numerical results again indicate that the non conservative shocks are properly computed by our scheme and with no numerical diffusion, compared to the usual (conservative) Roe's method. Note also that on Figure 7, the usual method of Roe applied to the conservative equations suffers from startup errors, leading to little wiggles.

At last, we conclude this section with a more realistic test case (Test 3) inspired from the first test case of [START_REF] Abgrall | Numerical approximation of a compressible multiphase system[END_REF] and involving a very large pressure jump in the initial data and across the shock wave. More precisely, we take (ρ, u, p) 0 (x) = (1185, 0, 2.0e11) if x < 0, (1185, 0, 1.0e5) if x > 0, where the density, velocity and pressure units are respectively kg/m 3 , m/s, and P a. The final time of simulation is t = 2e-5 and the mesh is made of 500 As expected, we observe on Figure 9 that the proposed method gives similar results compared to the usual method of Roe applied to the conservative system, but with less numerical diffusion across the shock.

Concluding remarks

A key property in the proof of Theorem 5.1 lies in the fact that the Roe scheme is able to provide an exact solution in the case of an isolated shock discontinuity. Note however that Roe's scheme is not entropic without correction for general systems. This therefore emphasizes the need for the development of such approximate Riemann solvers which are entropic and able to exactly reproduce isolated discontinuities. We refer the reader to [START_REF] Bouchut | Semi-discrete entropy satisfying approximate Riemann solvers. The case of the Suliciu relaxation approximation[END_REF], [START_REF] Chalons | Modified Suliciu relaxation system and exact resolution of isolated shock waves[END_REF] for contributions on this topic. The proposed numerical scheme is only statistically conservative on the conservative variables τ and u because of the random sampling, but it does not prevent the method from converging, numerically, to the right solution.

In one space dimension, the random sampling strategy is not mandatory. It could be easily replaced by considering moving meshes with non constant space steps, see for instance [START_REF] Zhong | Computational methods for propagating phase boundaries[END_REF], which would give a conservative scheme on a moving mesh. In several space dimensions, it is well-known from the work by Colella [START_REF] Collela | Glimm's method for gas dynamics[END_REF] that the random sampling strategy cannot be used as it stands if one simply considers directional splitting methods. An alternative is currently under preparation.

At last, the proposed method is currently being applied to more general systems, in particular really non conservative systems (by really it is meant that they do not admit a natural conservative form) of interest for various applications.

A The non conservative formulation in Eulerian coordinates

We briefly give in this section the basic material used in the previous section to obtain the numerical results in Eulerian coordinates. As we will see, the approach followed is a direct extension of the Lagrangian framework. Since the proposed averages are clearly consistant, it is also easy to ckeck that the second property holds true, namely that 2. for all w, B φ (w, w) = B(w).

In regards to the first property 1. for all w L and w R , B φ (w L , w R ) has 3 distinct eigenvalues, it is not always satisfied since the eigenvalues of B φ (w L , w R ) are shown to be

µ 1 = -→ u -γ(γ -1)ẽ, µ 2 = u, µ 3 = -→ u + γ(γ -1)ẽ.
As a consequence, while clearly µ 1 < µ 3 , it might happen that µ 1 = µ 2 or µ 2 = µ 3 since -→ u = u in general. Property 1. is thus satisfied under the condition µ 1 < µ 2 < µ 3 . In practice, this condition turned out to be satisfied for the test cases proposed in the numerical section.

To conclude, let us underline that the objective of this section, and more generally of this paper, was to show that a relevant projection step might give a numerical strategy to be able to properly compute the non conservative shocks and that, it can be considered in both Lagrangian or Eulerian coordinates. However and considering the present section in Eulerian coordinates, the proposed approach is certainly not the best one since Property 1. is only conditionnally satisfied. Considering a Lagrange-Projection strategy [START_REF] Godlewsky | Numerical approximation of hyperbolic systems of conservation laws[END_REF] (based on the Roe linearization introduced in the Lagrangian framework and followed by a transport step) or the development of a new approximate Riemann solver which is entropic and exact for isolated shock waves, might be more relevant. Going further into the details of this strategy will be the purpose of a forthcoming contribution, see the concluding section above.

Figure 3 :

 3 Figure 3: Test 2 in Lagrangian coordinates : τ (Left), p (Right) and u (Bottom). Exact solution (solid line), our scheme (cross), path-conservative scheme (square)

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: Test 2 in Lagrangian coordinates : τ (Left), p (Right) and u (Bottom). Exact solution (solid line), our scheme with 300 points (cross), our scheme with 800 points (square)

  First of all, the conservative form of the gas dynamics equations in Eulerian coordinates writes   ∂ t ρ + ∂ x ρu = 0, ∂ t ρu + ∂ x (ρu 2 + p) = 0, ∂ t ρE + ∂ x (ρEu + pu) = 0,where ρ is the density and the other quantities have already been defined before.The non conservative formulation under consideration in this section is the following natural extension of the Lagrangian framework, namely   ∂ t ρ + ∂ x ρu = 0, ∂ t ρu + ∂ x (ρu 2 + p) = 0, ∂ t ρe + ∂ x ρeu + p∂ x u = 0,where only the last equation has been changed. It can be given the following short form ∂ t w(x, t) + B(w)(x, t)∂ x w(x, t) = 0, where w = (ρ, ρu, ρe) and

  Test 2. The second test case is a given Riemann problem leading to three simple waves :

	8.5
	8																								1 1
	7.5
																									0.9 0.9
	7
																									0.8 0.8
	6.5
	6																								0.7 0.7
	5.5
																									0.6 0.6
	5
																									0.5 0.5
	4.5
	4																								0.4 0.4
	3.5
																									0.3 0.3
	3
																									0.2 0.2
	2.5
	2																								0.1 0.1
	1.5
	-0.1	0	1	0.15 0.1	0.2	0.25 0.2	0.3	0.35 0.3	0.4	0.4	0.45	0.5 0.5	0.55	0.6 0.6	0.65	0.7	0.7	0.75 0.8	0.8	0.85 0.9	0.9	1	0.95	1 1.1	1.05	-0.1 -0.1	0 0	0.1 0.1	0.2 0.2	0.3 0.3	0.4 0.4	0.5 0.5	0.6 0.6	0.7 0.7	0.8 0.8	0.9 0.9	1 1	1.1 1.1
							2.6 2.6
							2.4 2.4
							2.2 2.2
							2 2
							1.8 1.8
							1.6 1.6
							1.4 1.4
							1.2 1.2
							1 1
							0.8 0.8
							0.6 0.6
							0.4 0.4
							0.2 0.2
							0 0
							-0.2 -0.2
							-0.1 -0.1			0 0		0.1 0.1			0.2 0.2		0.3 0.3			0.4 0.4		0.5 0.5	0.6 0.6		0.7 0.7	0.8 0.8		0.9 0.9	1 1		1.1 1.1
																																	path-conservative
	scheme (square)
																									10

Figure

1

: Test 1 in Lagrangian coordinates : τ (Left), p (Right) and u (Bottom). Exact solution (solid line), our scheme (cross), Figure

2

: Test 1 in Lagrangian coordinates : τ (Left), p (Right) and u (Bottom). Exact solution (solid line), our scheme with 300 points (cross), our scheme with 800 points (square)

If we denote by w -and w + the left and right states of a shock discontinuity, and σ its speed of propagation, the Rankine-Hugoniot relations associated with the conservative formulation are (see for instance [START_REF] Godlewsky | Numerical approximation of hyperbolic systems of conservation laws[END_REF])

while the entropy criteria imposes in particular that (ρ + -ρ -) > 0 for a 1shock discontinuity associated with the first eigenvalue u -γ(γ -1)e and (ρ + -ρ -) < 0 for a 3-shock discontinuity associated with the third eigenvalue u + γ(γ -1)e. Recall that the second eigenvalue λ 2 = u is associated with a linearly degenerate characteristic field giving rise to contact discontinuities only. Like in the Lagrangian framework, the theory of Dal Maso, LeFloch and Murat [START_REF] Maso | Definition and weak stability of a non conservative product[END_REF] applies and leads to the generalized Rankine-Hugoniot relations

and the very simple choice of path given by

for all s ∈ [0, 1] allows to recover the original conservative Rankine-Hugoniot relations.

As a Roe linearization B φ , we take

where

and

It is interesting to note that B φ (w L , w R ) is not the matrix B applied to an average state since the averages involved in its second and third rows are not the same. However, such a choice is natural in order to get the third required property (which can be easily checked) of a Roe linearization, namely 3. for all w L and w R ,


Ahlem P Bouhlel

Christophe P Curti

Michèle P Bertrand

Patrice P Vanelle
email: patrice.vanelle@univ-amu.fr

Access to original spirocyclic derivatives via inter

Keywords:

Introduction

Mn(III)-based oxidative free-radical cyclizations have been extensively explored over the past 40 years. Indeed, Mn(OAc) 3 is an important tool in C-C bond formation in organic synthesis. 1 Several reactions have been conducted on ketoesters 2 like acetoacetate, 3 1,3-diketones, 2 -nitro ketones, 4 malonates, 2,5 -ketoenamines, 6 -ketosulfones, 7 -ketonitriles, 8 and -ketoamides. 9 For instance, Citterio and co-workers have explored additions to aromatic rings 10 and the subsequent formation of substituted tetralin. Snider and co-workers have also reported spirocyclic scaffold synthesis with this methodology. 11 Despite the existence of several synthetic methodologies to obtain spirocyclic compounds 12 the last decade has seen a large number of patents on spirocyclic scaffold synthesis and applications 13 illustrating the high level of interest in the unique spatial arrangement of spirocyclic rings in medicinal chemistry. 14 We report herein the synthesis of spirocyclic compounds via intermolecular radical addition mediated by Mn(OAc) 3 under mild conditions. This protocol was extended to the intramolecular Mn(OAc) 3 promoted multistep synthesis of substituted spirobenzophenanthridin-6(5H)-ones, spirolactones and tetrahydroquinolinones. These scaffolds have been extensively used for the design of many compounds exhibiting a wide range of biological activities (i.e. antitumor 15 or antiviral 16 activities).

Our study establishes an easily reproducible protocol allowing inter-or intramolecular spirocyclization of dicarbonyl structures.

Results and discussion

Intermolecular Mn(OAc) 3 -mediated reaction

Intermolecular Mn(OAc) 3 -mediated reaction between benzylmalonate and alkene was first reported by Citterio and co-workers in several studies. 10 With a 1,1-disubstituted alkene, 2,4,4-trimethylpent-1-ene, they described tetralin (63%) as the major product. Their reaction was conducted in acetic acid, at 60 °C, using a 1: 1.5: 2 molar ratio, corresponding respectively to substrate/ alkene/ Mn(III). 10a The use of a methylenecycloalkane as a 1,1-disubstituted alkene allowed efficient spirocyclization under mild conditions. Thus, the reaction yielded three different products, in proportions which varied with the reaction conditions: the elimination product 1, spirocyclic tetralin 2 and spirolactone 3 (Scheme 1).

A R T I C L E I N F O

A B S T R A C T Scheme 1. Radical oxidative cyclization via manganese acetate between benzyldiethylmalonate and methylenecyclohexane a reactant fractioned in five portions added successively b reaction conducted in ethyl alcohol for 72h.

As shown in Table 1, three products were formed in nearly equal yields in our first attempt (entry 1). Microwave irradiation at 80 o C improved slightly tetralin 2 and spirolactone 3 yields (entry 2), and at rt, none of these products was observed (entry 4). To investigate conditions that might favor spirocyclic tetralin formation, Mn(OAc) 3 was divided in 5 portions that were added successively (entry 3) in order to ensure, according to Snider, moderate oxidizing conditions. 11 Tetralin 2 was effectively obtained as the major product (52%) via radical aromatic substitution. Alkene fractioning had almost no effect on reaction yields (entry 5).

The formation of elimination product 1 and spirolactone 3 could be promoted under strong oxidative conditions. Adding copper acetate 17 (entries 7-8) led to an increase in elimination product 1 and spirolactone 3 yields, while these conditions drastically decreased the yield of compound 2. Replacing Mn(OAc) 3 by CAN also resulted in a decrease of product 2 yield (entry 9). These results can be explained by the reaction mechanism proposed in Scheme 2. 18 According to this mechanism, the reaction of Mn(OAc) 3 with diethylbenzylmalonate yields manganese(III)-enolate A. Then, radical B is formed. Addition of B to the alkene forms a carbon-carbon bond and radical C is obtained. There are two possible pathways. Under strong oxidative conditions, oxidation to carbocation D is promoted. This carbocation leads to product 1 via proton elimination or to spirolactone 3 via intramolecular cyclization-lactonization. 19 Under mild oxidation conditions, radical C is more slowly oxidized, which promotes the formation of tetralin 2. 18

Scheme 2. Mechanism for formation of elimination product 1, spirocyclic tetralin 2 and spirolactone 3

Intramolecular Mn(OAc) 3 -mediated reactions

We propose a multistep synthesis leading in parallel to three types of products: (i) spirobenzophenanthridin-6(5H)-one series 8a-c, (ii) tetrahydroquinolinone series 9a-c, (iii) and spirolactone attached to tetrahydroquinolinone, series 10a-c.

The required starting materials, ethyl 2-benzyl-3-(2-(cycloalkylidenemethyl)phenylamino)-3-oxopropanoate 7a-c, were synthesized in four steps as shown in Scheme 3.

First,

we synthesized 1-nitro-2-[(1nitrocycloalkyl)methyl]benzene 4a-c via the S RN 1 process. 20 This radical chain reaction is particularly efficient when pnitrobenzylchloride reacts with aliphatic and cyclic nitronate anions. Indeed, in the presence of 2-nitropropane anions, pnitrobenzylchloride led to 83-95% of C-alkylation. In contrast, o-nitrobenzylchloride led to only 46% of C-alkylation product. 21 Here, the S RN 1 reaction conditions had to change to ensure the reaction of o-nitrobenzylchloride with various nitrocycloalkanes. The synthesis was performed in two steps. First, nitrocycloalkane reacted with sodium hydride, under inert atmosphere and in DMSO 22 to give the corresponding nitronate anion. Then, 1-(chloromethyl)-2-nitrobenzene was added under light irradiation. Corresponding C-alkylation products were obtained in moderate to good yields (55-79%). Elimination products were also observed in very low yields (0-7%).

Compounds 4a-c, were then converted into the desired elimination products 1-(cycloalkylidenemethyl)-2nitrobenzene 5a-c, as a result of loss of nitrous acid from the C-alkylation products. Reaction led to good yields (61-94%).

The third step was nitro group reduction conducted with iron in acetic acid at 70 °C. 23 The 2-(cycloalkylidenemethyl)benzenamines 6a-c were obtained in good yields (75 to 80%).

Finally, the formation of amide linkage was carried out by reacting compounds 6a-c with ethyl 2-(chlorocarbonyl)-3phenylpropanoate. The reaction, conducted in 5 min at rt, led to ethyl 2-benzyl-3-(2-(cycloalkylidenemethyl)phenylamino)-3-oxopropanoate 7a-c in good yields (69-73%). -Ketoamides 7a-c, bearing a C-C double bond, were used as key substrates for intramolecular oxidative cyclization mediated by Mn(OAc) 3 .

i: NaH, DMSO, 18 h, rt, hii : NaH, DMSO, 2-3 h, rt iii: Fe, AcOH, 2-3h, 75 °C, iv:

, DMAP, toluene, 5 min, rt

Scheme 3. Synthesis of starting materials 7a-c

In acetic acid at 80 °C and under microwave irradiation, the first synthesis was conducted from ethyl 2-benzyl-3-[2-(cyclohexylidenemethyl)phenylamino]-3-oxopropanoate 7a. Mn(OAc) 3 , separated into five fractions, was added slowly every twenty min. The reaction led to three compounds: spirocyclic benzophenanthridin-6(5H)-one 8a (17%), 1,2,3,4tetrahydroquinolinone 9a (23%) and spirolactone 10a (19%) (Scheme 4, Table 2).

As shown in Scheme 4, the reaction of -ketoamide 7a and Mn(OAc) 3 affords radical A. Then, intramolecular radical addition to the double bond allows cyclization, and a new radical is formed as a mixture of two diastereomers (B and C). Bertrand and co-workers previously worked on Mn(III)mediated oxidative cyclizations of -diesters bearing a benzyl side chain. 24 Different mechanistic pathways may occur.

We postulate that when the carbon centered radical and the aromatic ring are cis (radical B), their proximity allows addition to the aromatic ring, and thus the formation of 8a. In contrast, when the carbon centered radical and the aromatic ring are trans (radical C), distance and steric hindrance slow down the rate of the aromatic substitution leading to transfused products. The proximity of the cylohexyl radical and the ester group promotes the formation of lactone 10a. Lactonization is likely to originate from the oxidation of the tertiary alkyl radical. The absence of any alkene resulting from oxidative elimination of radical C is very surprising, since the diastereomeric radical B was shown to lead to alkene 9a according to a similar pathway.

Under experimental conditions which promoted radical addition to the aromatic ring in the previously described intermolecular reactions (entry 2, Table 1), we observed only 19% of lactone 10a from radical C, and an overall 40% yield of products originating from radical B (Table 2). In the case of -diesters, Bertrand and co-workers reported that in the cyclization of 5-hexenyl radical, the diastereomer analogous to C was clearly preferred over radical B analogous, and that steric bulk of the substituents on the double bond enhanced this preference. The dramatic influence of the substituents on the double bond was demonstrated by recent mechanistic studies on Mn(OAc) 3 mediated cyclization of substituted allyl methyl--ketoesters. 25 Mn(III)-mediated oxidative cyclization was extended to cyclopentylidene and cycloheptylidene derivatives, 7b and 7c, respectively. Products such as spirobenzophenanthridin-6(5H)ones 8b-c, 1,2,3,4-tetrahydroquinolinones 9b-c and spirolactones 10b-c were obtained in moderate yields (11-29%). These results are reported in Table 2.

In the case of amides 7a-c bearing a disubstituted cycloalkyl scaffold the diastereoselectivity between radicals B and C was reversed as compared to that reported in the case of diesters. These differences probably result from the nature of the unsaturated tether, i.e., 5-hexenyl or 6-heptenyl radical intermediates (the latter being rigidified by the conjugate aromatic system). The fate of the 6-heptenyl radicals, formed from 7a-c was clearly probed, by the stereochemical assignment of products 8a, 9c, and 10a. The structure of compounds 8a, 9c and 10a was established by X-ray diffraction analysis (Figures 1, 2 and3). As product 9a is an oil, and NOE analysis was unsuccessful, its structure was deduced from the X-ray analysis of its cycloheptyl-analogous 9c. This crystal structure contains 4 conformers and an acetonitrile molecule in the asymmetric unit. Figure 2 represents the ORTEP view of only one of the 4 conformers, ORTEP views of all the asymmetric units of 9c is provided as supplementary data.

O Cl O O n = 1, 2, 3

Conclusion

An efficient and convenient radical oxidative cyclization initiated by manganese acetate between diethylmalonate and methylenecyclohexane was performed in this study. This reaction afforded three different types of products: an elimination product, a spirocyclic tetralin and spirolactone. By modifying the conditions, the spirocyclic compound could be obtained as the major product.

Then, extension to -ketoamide bearing a C-C double bond allowed us to obtain spirobenzophenanthridin-6(5H)-one 8a-c, 1,2,3,4-tetrahydroquinolinone derivatives 9a-c and spirolactone 10a-c. Stereocontrol should make it possible for this synthesis to serve as a key step in accessing various molecules with potential biological activities.

To further define the scope and limitations of our reaction and to extend our chemical library, applications of this reaction to other starting materials are in progress.

Experimental section

General

Microwave-assisted reactions were performed in a multimode microwave oven (ETHOS Synth Lab Station, Ethos start, Milestone Inc.). Melting points were determined with a B-540 Büchi melting point apparatus. 1 H NMR (200 MHz) and 13 C NMR (50 MHz) spectra were recorded on a Bruker ARX 200 spectrometer in CDCl 3 or DMSO-d 6 at the Service interuniversitaire de RMN de la Faculté de Pharmacie de Marseille. The 1 H chemical shifts are reported as parts per million downfield from tetramethylsilane (Me 4 Si), and the 13 C chemical shifts were referenced to the solvent peaks: CDCl 3 (76.9 ppm) or DMSO-d 6 (39.6 ppm). Absorptions are reported with the following notations: s, singlet; bs, broad singlet; d, doublet; t, triplet; q, quartet; m, a more complex multiplet or overlapping multiplets. Elemental analysis and mass spectra which were run on an API-QqToF mass spectrometer were carried out at the Spectropole de la Faculté des Sciences et Techniques de Saint-Jérôme. The following adsorbent was used for flash column chromatography: Silica Gel 60 (Merck, particle size 0.040-0.063 nm, 70-230 mesh ASTM). TLC were performed on 5 cm x 10 cm aluminium plates coated with silica gel 60 F-254 (Merck) in an appropriate solvent.

Protocol for Mn(OAc) 3 -mediated reaction leading to compounds 1-3 following conditions in entry 2 of Table 1

A solution of manganese(III) acetate dihydrate (1.68 mmol, 0.449 g) in 55 mL of glacial acetic acid was heated under microwave irradiation (200 W, 80 °C) for 15 min, until dissolution. Then, the reaction mixture was cooled down to 60 °C, and a solution of diethylbenzylmalonate (3.99 mmol, 1.0 g) and methylenecyclohexane (7.98 mmol, 0.767 g) in 5 mL of glacial acetic acid was added. The mixture was heated under microwave irradiation (200 W, 80 °C) for 20 min. Then, the reaction mixture was cooled down to 60 °C once more, and a second quantity of manganese(III) acetate dihydrate (1.68 mmol, 0.449 g) was added. The mixture was heated under microwave irradiation (200 W, 80 °C) for 20 min. The addition of manganese(III) acetate dihydrate (1.68 mmol, 0.449 g) was repeated three times in the same conditions every 20 min. successively. The reaction mixture was poured into 100 mL of cold water, and extracted with chloroform (3 x 70 mL). The organic extracts were collected, washed with saturated aqueous NaHCO 3 (3 x 50 mL) and brine (3 x 50 mL), dried over MgSO 4 , filtered, and concentrated under vacuum. The crude product was purified by silica gel chromatography with ethyl acetate/petroleum ether (0.5/9.5) to give compounds 1, 2, and 3.

Diethyl 2-benzyl-2-(cyclohexenylmethyl)malonate 1

Colorless oil; 1 H NMR (200 MHz, CDCl 3 )  H 1.20 (t, J=7. 1Hz, 6H, 2CH 3 ), 1.55-1.59 (m, 4H, 2CH 2 ), 1.90-2.00 (m, 4H, 2CH 2 ), 2.58 (s, 2H, CH 2 ), 3.26 (s, 2H, CH 2 ), 4.12 (q, J=7.1 Hz, 4H, 2CH 2 ,), 5.52 (s, 1H, 1CH), 7.11-7.24 (m, 5H, 5CH). 13

General procedure for the synthesis of C-alkylation products 4a-c

Into a two-necked flask containing 70 mL of DMSO, sodium hydride (0.372, 15.5 mmol) was added under inert atmosphere and at rt. Then, nitrocycloalkane (15.5 mmol) was added and the solution was stirred during 2-3 h. Under light stimulation, 1-(chloromethyl)-2-nitrobenzene (0.892 mg, 5.20 mmol) was added. The reaction mixture, maintained under inert atmosphere, was stirred at rt and monitored by TLC. When the starting material disappeared, the reaction was quenched with 50 mL of water at 0 °C. The reaction mixture was extracted with ethyl acetate (5 x 100 mL). The combined organic layers were washed with a solution of brine (5 x 200 mL), dried over MgSO 4 , filtered, and concentrated under vacuum. The crude product was purified by silica gel chromatography with dichloromethane/ petroleum ether/ diethylether (1/ 8.5/ 0.5) to give the corresponding C-alkylation products 4a-c. These products were recrystallized in isopropanol. 4.3.1. 1-Nitro-2-((1-nitrocyclohexyl)methyl)benzene 4a 26 White solid; mp 94 °C;

General procedure for the synthesis of elimination products 5a-c

To a solution of sodium hydride (0.202 g, 8.40 mmol) in 25 mL of DMSO, C-alkylation product 4 (2.80 mmol) was added under inert atmosphere. The reaction mixture was stirred at rt and monitored by TLC. When the starting material disappeared, the reaction was quenched with 25 mL of water at 0 °C. The reaction mixture was extracted with ethyl acetate ( 5x 50 mL). The combined organic layers were washed with a solution of brine (5 x 100 mL), dried over MgSO 4 , filtered, and concentrated under vacuum. The crude product was purified by silica gel chromatography with dichloromethane/ petroleum ether/ diethylether (1.5/ 8/ 0.5) to give the corresponding elimination products 5a-c.

1-(Cyclohexylidenemethyl)-2-nitrobenzene 5a 27

Yellow oil.

General procedure for the synthesis of 2-(cycloalkylidenemethyl)benzenamine 6a-c

To a solution of 1-(cycloalkylidenemethyl)-2-nitrobenzene 5ac (4 mmol) in 60 mL of acetic acid at 70°C, Fe (5.58 g, 100 mmol) was added. The resulting suspension was refluxed for 2-3 h. Then, the mixture was filtered on celite to remove Fe. The filtrate was extracted with dichloromethan (3 x 80 mL).

The combined organic layers were washed with a solution of brine (2 x 80 mL), dried over MgSO 4 , filtered, and concentrated under vacuum. The crude product was purified by silica gel chromatography with dichloromethane/ petroleum ether (3/ 7) to give the corresponding 2-(cycloalkylidenemethyl)benzenamine 6a-c. 4.5.1. 2-(Cyclohexylidenemethyl)benzenamine 6a 28 Yellow oil; 1 H NMR (200 MHz,CDCl 3 )  H 1.52-1.64 (m, 6H, 3CH 2 ), 2.17-2.32 (m, 4H, 2CH 2 ), 3.69 (bs, 2H, NH 2 ), 6.01 (s, 1H, CH), 6.68-6.76 (m, 2H, 2CH), 6.98-7.10 (m, 2H, 2CH). 13

General procedure for the synthesis of -ketoamide 7a-c

To a solution of ethyl 2-(chlorocarbonyl)-3-phenylpropanoate (0.520 mg, 2.16 mmol) in 10 mL of toluene were added 2-(cycloalkylidenemethyl)benzenamine 6a-c (1.44 mmol) and dimethylaminopyridine (0.352 mg, 2.88 mmol). The reaction mixture was stirred at rt for 5 min. Evaporation of the solvent left a yellow oil as crude product. Purification with dichloromathane by silica gel chromatography gave the corresponding -ketoamide 7a-c. 4.6.1. Ethyl 2-benzyl-3-(2-(cyclohexylidenemethyl) 378.2064. Found: 378.2066. 4.6.3. Ethyl 2-benzyl-3-(2-(cycloheptylidenemethyl)

General procedure for Mn(OAc) 3 -mediated reaction

A solution of manganese(III) acetate dihydrate (1.56 mmol,0.418 g) in 55 mL of glacial acetic acid was heated under microwave irradiation (200 W, 80 °C) for 15 min, until dissolution. Then, the reaction mixture was cooled down to 60 °C, and the corresponding -ketoamide 7a-c (3.71 mmol) in 5 mL of glacial acetic acid was added. The mixture was heated under microwave irradiation (200 W, 80 °C) for 20 min. Then, the reaction mixture was cooled down to 60 °C once more, and a second quantity of manganese(III) acetate dihydrate (1.56 mmol,0.418 g) was added. The mixture was heated under microwave irradiation (200 W, 80 °C) for 20 min. The addition of manganese(III) acetate dihydrate (1.56 mmol,0.418 g) was repeated three times in the same conditions every 20 min. successively. The reaction mixture was poured into 100 mL of cold water, and extracted with chloroform (3 x 70 mL). The organic extracts were collected, washed with saturated aqueous NaHCO 3 (3 x 50 mL) and brine (3 x 50 mL), dried over MgSO 4 , filtered, and concentrated under vacuum. The crude product was purified by silica gel chromatography with a gradient from ethyl acetate/petroleum ether (0.2/9.8) to ethyl acetate/petroleum ether (1/9) to give the corresponding products 8a-c, 9a-c, 10a-c. For solid products, a recrystallization was conducted in the appropriate solvent. 4.7.1. Ethyl 6-oxo-5,12a-dihydro-6Hspiro[benzo[j]phenanthridine-12,1'-cyclohexane]-6a(7H)carboxylate 8a White solid; recrystallized in diethyl ether; mp 255 °C; 1 H NMR (200 MHz, )  H 0.82 (t, J=7.0 Hz, 3H, CH 3 ), 1.12-1.33 (m, 2H, CH 2 ), 1.47-1.74 (m, 4H, 2CH 2 ), 1.88-2.10 (m, 4H, 2CH 2 ), 3.02-3.11 (m, 2H, CH 2 ), 3.79-3.98 (m, 3H, CH, CH 2 ), 6.93-6.98 (m, 2H, 2CH), 7.11-7.26 (m, 5H, 5CH), 7.60-

Scheme 4 .

 4 Scheme 4. Hypothesis of spirobenzophenanthridin-6(5H)-one 8a, tetrahydroquinoline 9a and spirolactone 10a formation

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. ORTEP view of one enantiomer of (±) 8aFigure2. ORTEP view of one enantiomer of (±) 9c

Table 1 .

 1 Survey of manganese acetate reaction conditions

	Entry	Alkene (equiv)	Mn(OAc)3 (equiv)	Oxidant (equiv)	Heating	Yield 1 (%)	Yield 2 (%)	Yield 3 (%)
	1	2	2.1	0	Classical, 60 °C	22	17	13
	2	2	2.1	0	MW, 80 °C	28	22	20
	3	2	2.1 a	0	MW, 80 °C	11	52	2
	4	2	2.1	0	none	0	0	0
	5	2 a	2.1	0	Classical, 60 °C	20	20	14
	6	2	3	0	MW, 80 °C	26	18	23
	7	2	3	Cu(OAc)2 (1)	MW, 80 °C	31	17	22
	8	2	6	Cu(OAc)2 (2)	MW, 80 °C	28	8	23
	9	2	0	CAN (2) b	none	27	0	19

Table 2 .

 2 Intramolecular oxidative radical cyclization of 7a, 7b, 7c via manganese acetate

  White solid; mp 75 °C;1 H NMR (200 MHz, CDCl 3 )  H 1.46 (s, 8H, 4CH 2 ), 1.74-1.82 (m, 2H, CH 2 ), 2.20-2.27 (m, 2H, CH 2 ), 3.59 (s, 2H, CH 2 ), 7.01 (d, J=6.7 Hz, 1H, CH), 7.34-7.42 (m, 2H, 2CH), 7.79 (d, J=7.2 Hz, 1H, CH). 13 C NMR (50 MHz, CDCl 3 )  C 22.4 (2CH 2 ), 29.4 (2CH 2 ), 36.2 (2CH 2 ), 41.1 (CH 2 ), 96.3 (C), 124.5 (CH), 128.3 (CH), 129.1 (C), 131.9 (CH), 132.7(CH), 150.1 (C). HMRS (ESI): m/z calcd for C 14 H 18 N 2 O 4 M+H + : 296.1605. Found: 296.1608.

	1 H NMR (200 MHz, CDCl 3 )  H 1.22-
	1.61 (m, 8H, 4CH 2 ), 2.34-2.41 (m, 2H, CH 2 ), 3.58 (s, 2H,
	CH 2 ), 7.08 (dd, J=7.4 Hz, J=1.6 Hz, 1H, CH), 7.39-7.55 (m, 2H, 2CH), 7.89 (dd, J=7.7 Hz, J=1.6 Hz, 1H, CH). 13 C NMR
	(50 MHz, CDCl 3 )  C 22.1 (2CH 2 ), 24.4 (CH 2 ), 33.6 (2CH 2 ),
	42.3 (CH 2 ), 92.5 (C), 124.9 (CH), 128.7 (CH), 128.8 (C),132.5
	(CH), 133.0 (CH), 150.2 (C).
	4.3.2. 1-Nitro-2-((1-nitrocyclopentyl)methyl)benzene 4b
	White solid; mp 51 °C; 1 H NMR (200 MHz, CDCl 3 )  H 1.70-
	1.87 (m, 6H, 3CH 2 ), 2.47-2.55 (m, 2H, CH 2 ), 3.78 (s, 2H,
	CH 2 ), 7.15 (dd, J=7.4 Hz, J=1.6 Hz, 1H, CH), 7.37-7.55 (m, 2H, 2CH), 7.88 (dd, J=7.8 Hz, J=1.6 Hz, 1H, CH). 13 C NMR
	(50 MHz, CDCl 3 )  C 23.2 (2CH 2 ), 36.5 (2CH 2 ), 39.2 (CH 2 ),

100.7 (C), 125.0 (CH), 128.5 (CH), 130.1 (C), 132.0 (CH), 133.1 (CH), 150.0(C). HMRS (ESI): m/z calcd for C 12 H 14 N 2 O 4 M+H + : 268.1292. Found: 268.1293.

4.3.3. 1-Nitro-1-(2-nitrobenzyl)

cycloheptane 4c

  1 H NMR (200 MHz, CDCl 3 )  H1.50-1.70 (m, 6H,

	3CH 2 ), 2.09-2.15 (m, 2H, CH 2 ), 2.26-2.32 (m, 2H, CH 2 ), 6.43
	(s, 1H, CH), 7.24-7.39 (m, 2H, 2CH), 7.35-7.57 (m, 1H, CH),
	7.91-7.95 (m, 1H, CH). 13 C NMR (50 MHz, CDCl 3 )  C 26.5
	(CH 2 ), 27.9 (CH 2 ), 28.4 (CH 2 ), 29.9 (CH 2 ), 37.2 (CH 2 ), 117.6
	(CH), 124.2 (CH), 127.0 (CH), 132.3 (CH), 132.4 (CH), 133.4
	(C), 145.6 (C), 148.8 (C).
	4.4.2. 1-(Cyclopentylidenemethyl)-2-nitrobenzene 5b 27
	Yellow oil. 1 H NMR (200 MHz, CDCl 3 )  H 1.67-1.75 (m, 4H,
	2CH 2 ), 2.34-2.41 (m, 2H, CH 2 ), 2.47-2.53 (m, 2H, CH 2 ), 6.61-
	6.63 (m, 1H, CH), 7.29-7.34 (m, 1H, CH), 7.45-7.56 (m, 2H,
	2CH), 7.86 (d, J=8.6 Hz, 1H, CH). 13 C NMR (50 MHz, CDCl 3 )
	 C 25.5 (CH 2 ), 26.7 (CH 2 ), 30.9 (CH 2 ), 35.0 (CH 2 ), 115.7 (CH),
	124.1 (CH), 126.5 (CH), 130.7 (CH), 132.2 (CH), 133.7 (C),
	148.2 (C), 151.1 (C).
	4.4.3. (2-Nitrobenzylidene)cycloheptane 5c
	Yellow oil; 1 H NMR (200 MHz, CDCl 3 )  H 1.53-1.72 (m, 8H,
	4CH 2 ), 2.21-2.26 (m, 2H, CH 2 ), 2.39-2.45 (m, 2H, CH 2 ), 6.46
	(s, 1H, CH), 7.30-7.39 (m, 2H, 2CH), 7.53 (td, J=7.5 Hz, J=1.2
	Hz, 1H, CH), 7.92 (d, J=8.2 Hz, 1H, CH). 13 C NMR (50 MHz,
	CDCl 3 )  C 27.1 (CH 2 ), 28.8 (CH 2 ), 28.9 (CH 2 ), 29.6 (CH 2 ),
	31.1 (CH 2 ), 37.7 (CH 2 ), 121.0 (CH), 124.1 (CH), 127.0 (CH),
	132.1 (CH), 132.3 (CH), 133.8 (C), 147.1 (C), 148.6 (C).
	HMRS (ESI): m/z calcd for C 14 H 17 NO 2 M+H + : 249.1598.
	Found: 249.1589.

  C NMR (50 MHz, CDCl 3 )  C 26.6 (CH 2 ), 28.0 (CH 2 ), 28.8

	(CH 2 ), 29.9 (CH 2 ), 37.1 (CH 2 ), 114.9 (CH), 117.6 (CH), 118.0
	(CH), 124.0 (C), 127.4 (CH) 130.2 (CH), 144.1 (C), 145.4 (C).
	4.5.2. 2-(Cyclopentylidenemethyl)benzenamine 6b 28
	Yellow oil; 1 H NMR (200 MHz, CDCl 3 )  H 1.67-1.74 (m, 4H,
	2CH 2 ), 2.38-2.48 (m, 4H, 2CH 2 ), 3.68 (s, 2H, NH 2 ), 6.22 (s,
	1H, CH), 6.67-6.78 (m, 2H, 2CH), 7.00-7.07 (m, 1H, CH),
	7.14 (d, J=7.5 Hz, 1H, CH). 13 C NMR (50 MHz, CDCl 3 )  C
	25.7 (CH 2 ), 26.6 (CH 2 ), 30.6 (CH 2 ), 34.5 (CH 2 ), 115.1 (CH),
	115.7 (CH), 118.2 (CH), 124.9 (C), 127.2 (CH), 129.0 (CH)
	143.6 (C), 148.6 (C).
	4.5.3. 2-(Cycloheptylidenemethyl)benzenamine 6c 28
	Yellow oil; 1 H NMR (200 MHz, CDCl 3 )  H 1.56-1.69 (m, 8H,
	4CH 2 ), 2.29-2.31 (m, 2H, CH 2 ), 2.40-2.46 (m, 2H, CH 2 ), 3.67
	(s, 2H, NH 2 ), 6.09 (s, 1H, CH), 6.68-6.77 (m, 2H, 2CH), 7.00-7.09 (m, 2H, 2CH). 13 C NMR (50 MHz, CDCl 3 )  C 27.2 (CH 2 ),
	29.3 (CH 2 ), 29.4 (CH 2 ), 29.7 (CH 2 ), 31.1 (CH 2 ), 37.8 (CH 2 ),
	114.8 (CH), 118.0 (CH), 121.0 (CH), 124.5 (C), 127.4 (CH),
	129.9 (CH), 143.9 (C), 147 (C).

  Yellow oil;1 H NMR (200 MHz, CDCl 3 )  H 1.17 (t, J=7.1 Hz, 3H, CH 3 ),1.64-1.75 (m, 4H, 2CH 2), 2.13-2.30 (m, 2H, CH 2 ), 2.41-2.57 (m, 2H, CH 2 ), 3.25 (dd, J=13.6 Hz, J=8.5 Hz, 1H, CH 2 ), 3.37 (dd, J=13.6 Hz, J=6.3 Hz, 1H, CH 2 ), 3.62 (dd, J=8.5 Hz, J=6.3 Hz, 1H, CH), 4.14 (q, J=7.1 Hz, 2H, CH 2 ), 6.10 (s, 1H, CH), 7.08-7.28 (m, 8H, 8CH), 8.13 (d, J=8.0 Hz, 1H, CH), 8.48 (s, 1H, NH). 13 C NMR (50 MHz, CDCl 3 )  C 13.9 (CH 3 ), 25.7 (CH 2 ), 26.5 (CH 2 ), 30.5 (CH 2 ), 34.4 (CH 2 ), 37.0 (CH 2 ), 55.8 (CH), 61.6 (CH 2 ), 114.8 (CH), 121.1 (CH), 124.1 (CH), 126.9 (CH), 127.0 (CH), 128.5 (2CH), 128.8 (CH), 128.9 (2CH), 134.6 (C), 137.6 (C), 151.4 (C), 165.5 (C), 171.4 (C). HMRS (ESI): m/z calcd for C 24 H 27 NO 3 M+H + :

	phenyl-
	amino)-3-oxopropanoate 7a
	Yellow oil; 1 H NMR (200 MHz, CDCl 3 )  H 1.17 (t, J=7.1 Hz,
	3H, CH 3 ), 1.46-1.70 (m, 6H, 3CH 2 ), 1.99-2.05 (m, 2H, CH 2 ),
	2.29-2.34 (m, 2H, CH 2 ), 3.26 (dd, J=13.7 Hz, J=8.7 Hz, 1H,
	CH 2 ), 3.38 (dd, J=13.7 Hz, J=6.3 Hz, 1H, CH 2 ), 3.62 (dd,
	J=8.7 Hz, J=6.3 Hz, 1H, CH), 4.14 (q, J=7.1 Hz, 2H, CH 2 ),
	5.94 (s, 1H, CH), 7.07-7.09 (m, 2H, 2CH), 7.21-7.32 (m, 6H,
	6CH), 8.24 (d, J=8.1 Hz, 1H, CH), 8.46 (s, 1H, NH). 13 C NMR
	(50 MHz, CDCl 3 )  C 13.9 (CH 3 ), 26.4 (CH 2 ), 27.7 (CH 2 ), 28.4
	(CH 2 ), 29.7 (CH 2 ), 36.8 (CH 2 ), 36.9 (CH 2 ), 55.9 (CH), 61.5
	(CH 2 ), 116.4 (CH), 120.3 (CH), 123.8 (CH), 126.8 (CH), 127.2
	(CH), 128.4 (C), 128.5 (2CH), 128.8 (2CH), 129.7 (CH), 135.4
	(C), 137.6 (C), 147.8 (C), 165.4 (C), 171.2 (C). HMRS (ESI):
	m/z calcd for C 25 H 29 NO 3 M+H + : 392.2220. Found: 392.2221.
	4.6.2. Ethyl 2-benzyl-3-(2-(cyclopentylidenemethyl)phenyl-
	amino)-3-oxopropanoate 7b

  Hz, J=8.9 Hz, 1H, CH 2 ), 3.35 (dd, J=13.7 Hz, J=6.1 Hz, 1H, CH 2 ), 3.59 (dd, J=8.9 Hz, J=6.1 Hz, 1H, CH), 4.14 (q, J=7.1 Hz, 2H, CH 2 ), 6.00 (s, 1H, CH), 7.06-7.09 (m, 2H, 2CH), 7.22-7.25 (m, 6H, 6CH), 8.23 (d, J=8.0 Hz, 1H, CH), 8.48 (s, 1H, NH). 13 C NMR (50 MHz, CDCl 3 )  C 13.9 (CH 3 ), 27.1 (CH 2 ), 28.9 (CH 2 ), 29.3 (CH 2 ), 29.6 (CH 2 ), 31.0 (CH 2 ), 37.1 (CH 2 ), 37.6 (CH 2 ), 56.0 (CH), 61.6 (CH 2 ), 119.8 (CH), 120.3 (CH), 123.9 (CH), 126.9 (CH), 127.3 (CH), 128.5 (2CH), 128.8 (C), 128.9 (2CH), 129.6 (CH), 135.3 (C), 137.6 (C), 149.9 (C), 165.4 (C), 171.3 (C). HMRS (ESI): m/z calcd for C 26 H 31 NO 3 M+H + : 406.2377. Found: 406.2370.

	phenyl-
	amino)-3-oxopropanoate 7c
	Yellow oil; 1 H NMR (200 MHz, CDCl 3 )  H 1.15 (t, J=7.1 Hz,
	3H, CH 3 ), 1.52-1.72 (m, 8H, 4CH 2 ), 2.15-2.17 (m, 2H, CH 2 ),
	2.42-2.47 (m, 2H, CH 2 ), 3.23 (dd, J=13.7

Acknowledgments

This work was supported by the Centre National de la Recherche Scientifique and the Université de la Méditerranée. We would like to express our thanks to V. Remusat for RMN spectra recording, M. Giorgi for X-ray diffraction and structure analysis, V. Monnier for mass spectra recording and M. Kaafarani and reviewers for useful discussions and advices.

G X-ray structure determination of compounds 8a, 9c and 10a

Crystal data for compound 8a: C 25 H 27 NO 3 , colourless prism (0.25x0.2x0.06 mm 3 ), M W = 389.48, monoclinic, space group P 21/c (T = 293 K), a = 14.2821( 3 Crystallographic data for the structure 8a, 9c and 10a have been deposited with the Cambridge Crystallographic Data Centre (CCDC) under the numbers 852539 for 8a, 864201 for 9c and 852540 for 10a. Copies of the data can be obtained, free of charge, on application to CCDC, 12 Union Road, Cambridge CB2 1EZ, UK (fax: +44 (0) 1223 336033 or e-mail: deposit@ccdc.cam.ac.uk).


Alex Meyebiyogo

Christophe Curti

Hussein El-Kashef

Omar Khoumeri

Thierry Terme

Patrice Vanelle
email: patrice.vanelle@univ-amu.fr

Mn(OAc) 3 catalyzed intermolecular oxidative peroxycyclization of naphthoquinone

a Manganese(III) acetate-mediated peroxycyclization between 2-hydroxy-3-methylnaphthoquinone and various alkenes was performed to obtain dihydronaphtho[2,3-c][1,2]dioxine-5,10(3H,10aH)-diones. The reactivity of symmetrical or unsymmetrical 1,1-disubstituted alkenes and monosubstituted alkenes allowed the synthesis of more than 50 original molecules. Focusing on the excellent reactivity of 2-hydroxy-3-methylnaphthoquinone, we describe the first example of Mn(OAc)3 reactivity with nitro-substituted alkenes. The scope, limitations and stereochemistry of the products synthesized are discussed. Starting from monosubstituted alkenes, the instability of a pair of diastereoisomers was observed, leading to ring opening.

Introduction

Manganese(III) acetate is a useful tool in organic chemistry,but remains underestimated in medicinal chemistry. Over the past four decades, the literature has contained accounts of its applications in C-C bond-forming reactions starting from various substrates. 1 Among these substrates, the reactivity of quinones was studied under anaerobic conditions with stoichiometric quantities of Mn(OAc) 3 .

2 Mn(OAc) 3 under aerobic conditions was also described as an efficient catalytic system, leading to C-C bond formation and oxygen insertion via a radical pathway, yielding 1,2-dioxane derivatives.

3 Saturated heterocyclic endoperoxides, such as 1,2-dioxane, were described as powerful antiparasitic compounds. In addition to the well-known commercial drug artemisinin, 4 trioxolanesarterolane and artefenomel, 5 a wide variety of 1,2-dioxanes showed potential antimalarial activity. 6 Quinone derivatives have important biological applications, mainly as anticancer 7 and antiparasitic 8 compounds.

The 2-hydroxy-3-alkylnaphthoquinone Lapachol derivatives antimalarial activity 9 encouraged us to seek an easily reproducible procedure to obtain chimeric 2-hydroxy-3-alkylnaphthoquinone / endoperoxidemolecules. Expanding on our previous work on the synthesis of antiparasitic compound 10 and Mn(OAc) 3 reactivity, 11 we present herein the first example of this original reactivity on 2-hydroxy-3methylnaphthoquinone with various alkenes, under Mn(OAc) 3 catalytic conditions.

Results and Discussion

First, best reaction conditions were studied with 1,1diphenylethylene (Scheme 1). According to previous Mn(OAc) 3 reactivity studies, good yields can be obtained with this alkene, as the intermediate benzylic tertiary radical is highly stabilized. 12 2-Hydroxynaphthoquinones were chosen both without and with a methyl in position 3. Results are summarized in Table 1.

Scheme 1.Mn(OAc)3 mediated cyclization of 2-hydroxynaphthoquinones with 1,1-diphenylethylene under aerobic conditions Please do not adjust margins Please do not adjust margins 1,1-diphenylethylene under aerobic conditions Entry [a] R Mn(OAc)3 (equiv.) Temperature (time) Product (yield) [b] 1 -H 2.5

65 o C (1 h) 1 (52%) 2 -H 2.5 RT (3 h) - 3 -H 0.4 RT (24 h) - 4 -H 0.4 65 o C (1 h) 1 (30%) 5 -CH3 2.5 65 o C 2 (63%) 6 -CH3 0.2 RT (1 h) 3 (95%) 7 -CH3 0.4 RT (0.5 h) 3 (90%)
[a] Reaction conditions: 2-hydroxynaphthoquinone (1 equiv), 1,1-diphenylethylene (0.5 equiv). [b] Isolated yields based on number of alkenes equiv.

Starting from 2-hydroxynaphthoquinone in the presence of Mn(OAc) 3 at 65 o C, dihydrofuronaphthoquinone1 was obtained in moderate yields (Entry 1) according to a previously reported mechanism. 2 We obtained lower yields (52%) than those reported with similar 1,1-diaromatic alkenes, perhaps due to the presence of oxygen under our conditions. Catalytic quantities of Mn(OAc) 3 only decreased yields without any oxygen insertion (Entry 4), and lower temperature (RT) left the unchanged starting material (Entries 2,3). With 2-hydroxy-3-methylnaphthoquinone in the presence of 2.5 equiv. of Mn(OAc) 3 at 65 o C, product 2 was formed, without traces of 1,2-dioxane ring formation (Entry 5). This structure was obtained via the same mechanism observed for product 1, but the presence of a methyl group in position 3 did not allow cyclization. Instead, acetate reacted with the intermediate carbocation to form product 2. Reactivity allowing the purification of product 3 in very good yields was finally observed with catalytic quantities of Mn(OAc) 3 at room temperature in only 1 h (Entry 6). When Mn(OAc) 3 quantities were doubled to 0.4 equiv, time reaction was decreased to 30 min with a moderate yield decrease (Entry 7). Temperature was previously reported to influence the Mn(OAc) 3 reactivity. 3a,12 At high temperature, under anaerobic conditions, a dihydrofuran ring is obtained, and at room temperature under aerobic conditions, oxygen trapping led to a 1.2-dioxane ring. Catalytic quantities of Mn(OAc) 3 are sufficient for this mechanism, summarized in scheme 2.

Scheme 2. Naphthoquinone / Mn(OAc)3 proposed reactivity mechanism.

Methyl group substitution in position 3 of naphthoquinone highly influenced Mn(OAc) 3 reactivity, appearing to be an important parameter leading to the formation of the 1.2-dioxane ring of product 3. The influence of the alkyl groups on Mn(OAc) 3 reactivity was previously studied, for example with -alkyl--keto esters versus-unsubstituted--ketoesters as summarized in Scheme 3.

This journal is © The Royal Society of Chemistry 20xx RSC Advances,2016,00,123Please do not adjust margins Please do not adjust margins The reaction was extended to various substrates, starting from symmetrical alkenes (Scheme 4). Results are summarized in Table 2. 1 [b] -Ph 3 (90%)

2 [b] -4-Cl-Ph 4 (86%)

3 [b] -4-F-Ph 5 (79%)

4 [c] -CH2-CH3 6 (54%)

5 [c] -(CH2)5-7 (40%)

[a] Isolated yields based on number of alkenes equiv. [b] Reaction conditions: 2-hydroxy-3methylnaphthoquinone (1 equiv), alkene (0.5 equiv). [c] Reaction conditions: 2-hydroxy-3methylnaphthoquinone (1 equiv), alkene (1 equiv).

Cis-substitution was observed for every product obtained from aromatic and aliphatic symmetrical alkenes. Yields obtained ranged from excellent with aromatic alkenes (Entries 1-3), even when they carried electron-withdrawing substituents such as -Cl or -F, to moderate (Entries 4-5) with aliphatic alkenes. This can be explained by the stability of intermediate radical formed during the reaction just after the C-C bond formation. For each type of alkene, a tertiary radical was formed, sufficiently stabilized to react with oxygen, but for aromatic alkenes the stability of this tertiary benzylic radical was increased by resonance.

From these promising results, we extended this reactivity to 1,1disubstituted unsymmetrical alkenes, in order to generate a new asymmetric center (Scheme 5). Results are summarized in Table 3. 3. Unsymmetrical 1,1-disubstituted alkenes reactivity.

Entry [a] R1 R2 Products (yields; diastereoisomeric ratio) [b] 1 [a] Reaction conditions: 2-hydroxy-3-methylnaphthoquinone (1 equiv), alkene (0.5 equiv). [b] Isolated yields based on number of alkenes equiv.

As expected, the reactivity of 2-hydroxy-3-methylnaphthoquinone and unsymmetrical alkenes led to a mixture of diastereoisomers. Two fractions were easily separable by column chromatography.

The first fraction corresponded to products with the R 2 and 4methyl group cis-substituted (8-16), and the second fraction to products with the R 2 and 4-methyl group trans-substituted (8'-16').

These results are evidence of the excellent reactivity of 2-hydroxy-3-methylnaphthoquinone for Mn(OAc) 3 C-C bond forming reactions, even with deactivated alkenes such as nitrile-(Entry 7) or nitro-(Entries 1, 8, 9) substituted alkenes. Mn(OAc) 3 reactivity of the double bond decreases with aromatic substitution of electronwithdrawing substituents. To our knowledge, this is the first example of a C-C bond forming reaction mediated with Mn(OAc) 3 and nitro-substituted alkenes. We did not observe reactivity for 1nitro-2-(prop-1-en-2-yl)benzene (Entry 10), probably because of steric hindrance due to the proximity of nitro and methyl groups with a double bond.Methoxy-substituted alkenes allowed the synthesis of products 12 / 12' (Entry 4) in lower yields than other alkenes. This phenomenon was already observed with methoxysubstituted alkenes, and an interaction between oxygen and Mn(OAc) 3 was hypothesized.

11d

Under conditions involving stoichiometric quantities of manganese, this limitation dealt by increasing manganese equivalents, but it remains problematic with catalytic quantities. This journal is © The Royal Society of Chemistry 20xx

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The structure of products 15/ 15' was confirmed by X-ray crystallography and an ORTEP view is reported in Figure 2. As previously described for 1,1disubstituted symmetrical alkenes, 10-hydroxyl and 4-methyl groups were cis-substituted. Moreover, for product 15, the R 2 substituent and 4-methyl group were also cissubstituted, while they were trans-substituted for product 15'. We noticed a slight stereoselectivity, with a diastereoisomeric excess for products cis substituted with 10-hydroxyl and R 2 substituents. Similar selectivity was previously described and discussed for radical induced peroxycyclization. 15 Next, given the excellent reactivity of 2-hydroxy-3methylnaphtoquinone, we extended reactivity to monosubstituted terminal alkenes (Scheme 6). Several vinyl derivatives reacted with naphthoquinone in moderate to excellent yields. Results are summarized in Table 4. Entry [a] R Diastereoisomer Products (yield; diastereoisomeric Ring opening products (yield) [b] ratio) [b] [a] Reaction conditions: 2-hydroxy-3-methylnaphthoquinone (1 equiv.), alkene (0.5 equiv.). [b] Isolated yields based on number of alkenes equiv. [c] Reaction was left under stirring for 2 hours after starting material consumption.

Starting from styrene or monosubstituted alkenes bearing electrondonating substituents, diastereoisomer products were synthesized in good yields in a similar way to that described previously. Surprisingly, stereoselectivity was not observed for styrene (Entry 1). Para-substitution for alkenes led to better yields than meta-or ortho-substitution (Entries 2, 3 and 5). When alkenes bearing electron-withdrawing substituents were used as starting material, only one diastereoisomeric product was purified and a major product identified as a ring opening structure (Entries 6-12). To confirm whether this new product came from a secondary reaction or from the degradation of peroxycyclized products, a reaction that only yielded to diastereoisomeric products was left under stirring for 3 hours (Entry 4). Similar formation of ring opening product 19'' was observed, clearly indicating that diastereoisomer19' was totally cleaved.

Decomposition of a 1,2-dioxane ring to afford -diketones was previously described. 16 Thus, the formation of the ring opening structure could be explained by the acid-catalyzed rearrangement of the less stable diastereoisomer (Scheme 7). Electron-withdrawing substituents on the benzene ring decrease the stability of the 1,2-dioxane ring by their influence on O-O bond stability and on benzylic hydrogen acidity. Therefore, spontaneous decomposition of peroxycyclized products could occur.

This journal is © The Royal Society of Chemistry 20xx RSC Advances, 2016, 00, 1-3 | 5

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The structure of products 27/ 27'' was confirmed by X-ray crystallography and an ORTEP view is shown in Figure 3. Peroxycyclizedproductstrans-substitutedwith 10-hydroxyl and R substituentappearedtobelessstablethancis-substituted ones. This differenceinstabilitymay also explainthestereoselectivityobservedduringcyclization.

Moreover, startingfrommonosubstitutedalkenes, when R bears an electronwithdrawingsubstituent, transsubstituteddiastereoisomerswereprobablyspontaneouslycleavedver yclosetotheirformation. On theotherhand, if R issubstitutedby an electron-donatinggroup, theincreasedreaction time couldleadtothe same decomposition on trans-substituteddiastereoisomers.

Conclusions

A new protocol of manganese(III) acetate catalyzed peroxycyclization of 2-hydroxy-3-methylnaphthoquinone was established and extended to a wide variety of alkenes. The high reactivity of quinones allowed deactivated alkenes to react with good yields. Thus, we report herein the first example of nitrosubstituted alkene Mn(OAc) 3 radical reactivity. Symmetrical or unsymmetrical 1,1-disubstituted alkenes and monosubstituted alkenes allowed the synthesis of 26 original molecules as a mixture of diastereoisomers or enantiomers. Differences in stability between diastereoisomers could explain stereoselectivity. Moreover, reactivity of monosubstituted alkenes bearing an electron-withdrawing group led to ring opening of the less stable diastereoisomers.

Experimental Section

Typical procedure for the synthesis of 3 A solution of 2-hydroxy-3-methyl-1,4-naphthoquinone (1.06 mmol, 200 mg, 1 equiv.) in 25 mL of glacial acetic acid was stirred at room temperature in an open vessel. 1.1-Diphenylethylene (0.53 mmol, 96 mg, 0.5 equiv.) and manganese(III) acetate dihydrate (0.21 mmol, 57 mg, 0.2 equiv.) were added and the reactionwas controlled by TLC until consumption of alkene. After 1 h, the reaction mixture was poured into 60 mL of cold water and extracted with dichloromethane (3 × 20 mL). The organic extracts were collected and dried (Na 2 SO 4 ). Solvent was evaporated under reduced pressure and crude product was purified by column chromatography (CH 2 Cl 2 /petroleum ether, 70/30), and the product obtained was recrystallized to afford product 3 (202 mg, 95%) as a white solid, mp = 167-168 o C (petroleumether/CH 2 Cl 2 , 9/1). 1 H NMR (250 MHz, CDCl 3 ): δ= 7.96-7.99 (m, 1H, Ar-H), 7.53-7.77 (m, 3H, Ar-H), 7.23-7.40 (m, 7H, Ar-H), 7.11-7.14 (m, 3H, Ar-H), 5.04 (s, 1H, OH), 3.68 (d, J = 13.6 Hz, 1H, CH), 2.88 (d, J = 13.6 Hz, 1H, CH), 1.25 (s, 3H, CH 3 ).

Figure 1 .

 1 Figure 1.ORTEP view of product 3.

  Scheme 4.Mn(OAc)3catalyzed cyclization of 2-hydroxy-3methylnaphthoquinone and symmetrical alkenes under aerobic conditions

Scheme 5 .

 5 Scheme5.Mn(OAc)3catalyzed cyclization of 2-hydroxy-3methylnaphthoquinone and unsymmetrical alkenes under aerobic conditions

Figure 2 .

 2 Figure 2.ORTEP view of products 16 / 16'.

Figure 2 .

 2 Figure 2. ORTEP view of products 15 / 15'.

  Scheme6.Mn(OAc)3 catalyzed cyclization of 2-hydroxy-3methylnaphthoquinone and monosubstituted alkenes under aerobic conditions

Scheme 7 .

 7 Scheme 7.Acid-catalyzed decomposition of peroxycyclizeddiastereoisomer.

Figure 3 .

 3 Figure 3. ORTEP view of products 27 / 27''.

13 C

 13 NMR (62.5MHz, CDCl 3 ): δ= 195.5 (C), 190.5 (C), 142.8 (C), 140.4 (C), 135.6 (CH), 134.0 (CH), 133.9 (C), 131.1 (C), 128.4 (2CH), 128.3 (CH), 127.7 (2CH), 127.5 (2CH), 127.4 (CH), 126.8 (CH), 126.7 (CH), 126.4 (2CH), 98.8 (C), 86.2 (C), 49.0 (C), 37.7 (CH 2 ), 23.8 (CH 3 ). HRMS (ESI): m/z [M + Na]+ calcdfor C 25 H 20 O 5 : 423.1203 ; found : 423.1204.

Table 1 .Mn(OAc)3 mediated

 1

cyclization of 2-hydroxynaphthoquinones and This journal is © The Royal Society of Chemistry 20xx

Table 2 .

 2 Symmetrical alkenes reactivity.

	Entry	R	Product (yield) [a]

Scheme 3. Mn(OAc)3 reactivity with  -ketoesters.13 2-Hydroxynaphthoquinones are already enolized and the Mn(III) enolate should form readily. Moreover, a methyl group in position 3 should facilitate oxidation, forming a stabilized radical. Without this methyl group, 2-hydroxynaphthoquinone reactivity would probably not be sufficient to initiate the mechanism with catalytic quantities of Mn(OAc) 3 and / or room temperature.The structure of product 3 was confirmed by X-ray crystallography and an ORTEP view is reported to Figure1. This crystallographic analysis confirmed that oxidative cyclization, theoretically possible on carbon 2 or on carbon 4 of naphthoquinones, only took place on carbon 2, probably because of its greater electrophilic nature. Crystallographic analysis also showed that 10-hydroxyl and 4methyl groups of product 3 were cis-substituted. A hydroxyl group occupies the axial position, due to the anomeric effect.

 14 During the intramolecular cyclization (Scheme 2), nucleophilic addition on carbonyl only occurs on the opposite side of the methyl group, yielding to a mixture of two cis-enantiomers.

Acknowledgements

This work was supported by the CNRS and Aix-Marseille University. The authors thank V. Remusat for the NMR spectra recording and V. Monnier for the mass spectra recording.

Notes and references