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1 Manuscript submitted to AIMS Journals Volume X, Number X, XX 2X Website: pp. X XX TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS hal-8732, version - 4 Oct 23 Ibrahima Faye Université de Bambey,UFR S.A.T.I.C, BP 3 Bambey (Sénégal, Ecole Doctorale de Mathématiques et Informatique. Laboratoire de Mathématiques de la Décision et d Analyse Numérique (L.M.D.A.N F.A.S.E.G/F.S.T. Emmanuel Frénod and Diaraf Seck Université Européenne de Bretagne, LMBA(UMR625 Université de Bretagne-Sud, Centre Yves Coppens, Campus de Tohannic, F-567, Vannes Cedex, France ET Projet INRIA Calvi, Université de Strasbourg, IRMA, 7 rue René Descartes, F-6784 Strasbourg Cedex, France Université Cheikh Anta Diop de Dakar, BP 6889 Dakar Fann, Ecole Doctorale de Mathématiques et Informatique. Laboratoire de Mathématiques de la Décision et d Analyse Numérique (L.M.D.A.N F.A.S.E.G/F.S.T. ET UMMISCO, UMI 29, IRD, France (Communicated by the associate editor name Abstract. In this paper we consider the model built in [3] for short term dynamics of dunes in tidal area. We construct a Two-Scale Numerical Method based on the fact that the solution of the equation which has oscillations Two-Scale converges to the solution of a well-posed problem. This numerical method uses on Fourier series.. Introduction. This paper deals with numerical simulations of sand transport problems. Its goal is to build a Two-Scale Numerical Method to simulate dynamics of dunes in tidal area. This paper enters a work program concerning the development of Two-Scale Numerical Methods to solve PDEs with oscillatory singular perturbations linked with physical phenomena. In Ailliot, Frénod and Monbet [2], such a method is used to manage the tide oscillation for long term drift forecast of objects in coastal ocean waters. Frénod, Mouton and Sonnendrücker [5] made simulations of the D Euler equation using a Two-Scale Numerical Method. In Frénod, Salvarani and Sonnendrücker [6], such a method is used to simulate a charged particle beam in a periodic focusing channel. Mouton [9, ] developped a Two-Scale Semi Lagrangian Method for beam and plasma applications. We consider the following model, valid for short-term dynamics of dunes, built and studied in [3]: z (A z = C, z t= = z, (. 2 Mathematics Subject Classification. Primary: 35K65, 35B25, 35B ; Secondary: 92F5, 86A6. Key words and phrases. Homogenization, Asymptotic Analysis, Asymptotic Expansion, Long Time Behavior, Dune and Megaripple Morphodynamics, Modeling Coastal Zone Phenomena, Numerical Simulation. This work is supported by NLAGA(Non Linear Analysis, Geometry and Application Project.

2 2 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK where z = z (t, x is the dimensionless seabed altitude. For a given T, t (, T stands for the dimensionless time and x T 2, T 2 being the two dimensional torus R 2 /Z 2, stands for the dimensionless position and A, C are given by A (t, x = à (t, x + à (t, x, (.2 and where, for three positive constants a, b and c, C (t, x = C (t, x + C (t, x, (.3 à (t, x = Ã(t, t, x = a ga( U(t, t, x, (.4 hal-8732, version - 4 Oct 23 and with C (t, x = C(t, t, x = c gc( U(t, t, x U(t, t, x U(t, t, (.5, x à (t, x = Ã(t, t, x, C (t, x = C (t, t, x, (.6 à (t, θ, x = abm(t, θ, x g a( U(t, θ, x and C (t, θ, x = cbm(t, θ, x g c( U(t, θ, x U(t, θ, x U(t, θ, x. (.7 U and M are the dimensionless water velocity and height. The small parameter involved in the model is the ratio between the main tide period = 3 hours and an observation ω time which is about three months i.e. = t ω = 2. The following hypotheses on g a, g c, U and M given in (.8 and (.9 are technical assumptions and are needed to prove Theorem.. Functions g a and g c are regular functions on R + and satisfy g a g c, g c( = g c ( =, d, sup u R + g a(u + sup u R + g a(u d, sup u R + g c(u + sup u R + g c(u d, U thr, G thr >, such that u U thr = g a(u G thr. Functions U and M are regular and satisfy: θ (U, M is periodic of period, U, U U,, U, θ (.8 M, M, M, M are bounded by d, θ (t, θ, x R + R T 2, U(t, θ, x U thr = ( U (t, θ, x =, U(t, θ, x =, M (t, θ, x =, and M(t, θ, x =, θ α < θ ω [, ] such that θ [θ α, θ ω] = U(t, θ, x U thr. To develop the Two-Scale Numerical Method, we use that in [3] we proved that under assumptions (.8 and (.9 the solution z of (. exists, is unique and moreover asymptotically behaves, as, the way given by the following theorem. Theorem.. Under assumptions (.8 and (.9, for any T, not depending on, the sequence (z of solutions to (., with coefficients given by (.2 coupled with (.4 and (.3, (.5 and (.6, Two-Scale converges to the profile Z L ([, T ], L # (R, L2 (T 2 solution to where à and C are given by (.9 Z θ (à Z = C, (. Ã(t, θ, x = a g a( U(t, θ, x and C(t, θ, x = c g c( U(t, θ, x U(t, θ, x U(t, θ, x. (.

3 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS 3 Futhermore, if the supplementary assumption is done, we have U thr =, (.2 Ã(t, θ, x G thr for any t, θ, x [, T ] R T 2, (.3 and, defining Z = Z (t, x = Z(t, t, x, the following estimate holds for z Z where α is a constant not depending on. Because of assumptions (.8 and (.9, z Z L ([,T,L 2 (T 2 α, (.4 Ã, C, Ã, C, Ã, Ã, C, and C are regular and bounded. (.5 hal-8732, version - 4 Oct Two-Scale Numerical Method Building. In this section, we develop the Two-Scale Numerical Method in order to approach the solution z of (.. The idea is to get a good approximation of z (t, x seeing Theorem. content as z (t, x Z(t, t, x. The strategy is to consider a Fourier expansion of Z solution to (.. In this equation, t is only a parameter. The Fourier expansion of Z is given as follows: Z(t, θ, x = Z (t e 2iπ(lθ+mx +nx 2, (2. where Z (t, l =,, 2,..., m =,, 2,..., n =,, 2,..., are the unknown complex coefficients of the Fourier expansion of Z. Using (2., the Fourier expansion of Z is given by θ Z (t, θ, x = 2iπ l Z (t e 2iπ(lθ+mx +nx 2. (2.2 θ To obtain the system satisfied by the Fourier expansion (2. of Z, it is necessary to compute the Fourier expansions of (à Z and C. As (à Z = à Z + à Z, let à (t e 2iπ(lθ+mx +nx 2, (2.3 and be respectively the Fourier expansions of à and expansions of Z and Z are respectively given by ( m 2iπ n and à grad (t e2iπ(lθ+mx +nx 2, (2.4 ( m Ã, where Ãgrad (t = 2iπà n and then the Fourier Z (t e 2iπ(lθ+mx +nx 2, (2.5 4π 2 (m 2 + n 2 Z (t e 2iπ(lθ+mx +nx 2. (2.6 In the same way the Fourier expansion of C is given by C e 2iπ(lθ+mx +nx 2. (2.7 Using (2., (2.2, (2.3, (2.4, (2.5, (2.6 and (2.7, equation (. becomes 2iπ l Z (t e 2iπ(lθ+mx +nx 2 ( ( à grad (t ( e2iπ(lθ+mx +nx 2 m 2iπ Z n (t e 2iπ(lθ+mx +nx 2 ( ( + à (t e 2iπ(lθ+mx +nx 2 4π 2 (m 2 + n 2 Z (t e 2iπ(lθ+mx +nx 2 = C (t e 2iπ(lθ+mx +nx 2, (2.8

4 4 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK which gives after identification, the following algebraic system for (Z : 2iπ l Z (t i,j,k 2iπÃgrad i,j,k (t ( m j n k Z l i,m j,n k (t +4π 2 i,j,k à i,j,k (t((m j 2 + (n k 2 Z l i,m j,n k (t = C (t. (2.9 hal-8732, version - 4 Oct 23 In formula (2., the integers m, n and l vary from to +. But in practice, we will consider the truncated Fourier series of order P N defined by Z P (t, θ, x = Z (t e 2iπ(lθ+mx +nx 2. (2. Using (2., formula (2.9 becomes: 2iπ l Z (t +4π 2 3. Convergence result. i P, j P, k P l P, m P, n P i P, j P, k P 2iπÃgrad i,j,k (t ( m j n k Z l i,m j,n k (t à i,j,k (t((m j 2 + (n k 2 Z l i,m j,n k (t = C (t. (2. Proof. of Theorem.. For self-containedness, we recall the proof of Theorem.. Firstly, we obtain an estimate leading to that z is bounded in L ((, T ; L 2 (T 2. Secondly, defining test function ψ (t, x = ψ(t, t, x for any ψ(t, θ, x, regular with a compact support over [, T T 2 and -periodic in θ, multiplying (. by ψ and integrating over [, T T 2 gives T z T 2 ψ dtdx T (A z ψ dtdx = T C ψ dtdx. (3. T 2 T 2 Then integrating by parts in the first integral over [, T and using the Green formula in T 2 in the second integral we have T ψ z (xψ(,, xdx T 2 T 2 z dtdx + T A z ψ dtdx = T C ψ dtdx. (3.2 T 2 T 2 Again using the Green formula in the third integral we obtain T ψ z (xψ(,, x dx T 2 T 2 z dtdx T z (A ψ dtdx = T C ψ dtdx. (3.3 T 2 T 2 But where then we have ψ = ( ψ + ( ψ (t, x = ψ (t, t, x and ( ψ, (3.4 θ ( ψ (t, x = ψ θ θ (t, t, x, (3.5 T (( ψ z + ( ψ + T 2 θ (A ψ dxdt + T C ψ dtdx = z (xψ(,, x dx. (3.6 T 2 T 2 Using the Two-Scale convergence due to Nguetseng [] and Allaire [] (see also Frénod Raviart and Sonnendrücker [7], since z is bounded in L ([, T, L 2 (T 2, there exists a profile Z(t, θ, x, periodic of period with respect to θ, such that for all ψ(t, θ, x, regular with a compact support with respect to (t, x and -periodic with respect to θ, we have T T z ψ dtdx Zψ dθdtdx, as tends to zero, (3.7 T 2 T 2

5 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS 5 hal-8732, version - 4 Oct 23 for a subsequence extracted from (z. Multiplying (3.6 by, passing to the limit as and using (3.7 we have T Z ψ T T dθdtdx + lim z (A ψ dtdx = lim C ψ dtdx, (3.8 T 2 θ T 2 T 2 for an extracted subsequence. As A and C are bounded and ψ is a regular function, A ψ and ψ can be considered as test functions. Using (3.7 we have T T z (A ψ dtdx Z (à ψ dθdtdx, (3.9 T 2 T 2 and T T C ψ dtdx Two-Scale converges to C ψ dθdtdx. (3. T 2 T 2 Passing to the limit as we obtain from (3.8 a weak formulation of the equation (. satisfied by Z. Using (.2 and (.3 equation (. becomes For Z, we have where Using (., Z is solution to Formulas (3. and (3.4 give z (à z = C + (à z + C. (3. Z = ( Z + ( Z (t, x = Z (t, t, x and (z Z ( Z, (3.2 θ ( Z (t, x = Z θ θ (t, t, x. (3.3 Z (à Z = ( C Z +. (3.4 (à ( (z Z = C Z + + (à z. (3.5 Multiplying equation (3.5 by and using the fact that z = z Z + Z in the right hand side of equation (3.5, z Z is solution to: ( z Z ( (à + à ( z Z = ( C + ( Z + (à Z. (3.6 Our aim here is to prove that z Z is bounded by a constant α not depending on. For this let us use that Ã, Ã, C and C are regular and bounded coefficients (see (.5 and that à G thr (see (.3. Hence, C is bounded, (à Z is also bounded. Since Z is solution to (3.4, Z satisfies the following equation ( Z θ ( à Z ( C à = + Z. (3.7 Equation (3.7 is linear with regular and bounded coefficients. Using a result of Ladyzenskaja, Solonnikov and Ural Ceva [8], Z is regular and bounded and so the coefficients of equations (3.6 are regular and bounded. Then, ( using the same arguments as in the proof of Theorem. in [3] we obtain that z Z is bounded. To determine the (à value of the constant α, we proceed in the same way as in the proof of Theorem 3.6 of [3]. Since the coefficients, Ã, C and C, C, (à Z, and Z are bounded by constants, let β denotes the maximum between all these constants. Then we use the same argument as in the proof of Theorems. and 3.6 and we get: z Z z β + β ( Z(,, β T. (3.8 L ([,T,L 2 (T 2 G thr

6 6 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK Theorem 3.. Let be a positive real, z be the solution to (., Z P be the truncated Fourier series (defined by (2. of Z solution to (. and ZP defined by Z P (t, x = Z P (t, t, x. Then, under assumptions (.8, (.9 and (.2, z ZP satisfies the following estimate: z Z P L ([,T,L 2 (T 2 z ( Z(,, 2 β + β 3 G thr + 2β T + f(p, (3.9 where f is a non-negative function of P not depending on and satisfying lim P + f(p =. Proof. We can write : z Z P L ([,T,L 2 (T 2 = z Z + Z Z p L ([,T,L 2 (T 2 z Z L ([,T,L 2 (T 2 + Z Z p L ([,T,L 2 (T 2. (3.2 Using (3.8, the first term in the right hand side of (3.2 is bounded by z Z L ([,T,L 2 (T 2 z ( Z(,, 2 β + β 3 G thr + 2βT. (3.2 hal-8732, version - 4 Oct 23 For the second term of (3.2, using classical results of Fourier series theory, since Z Z P is nothing but the rest of the Fourier series of order P of Z and since Z is regular (because it is the solution of (. which has regular coefficients, the non-negative function f satisfying lim P + f(p = such that exists. From this last inequality, follows and coupling this with (3.2 and (3.2 gives inequality ( Numerical illustration of Theorem 3.. Z Z p L ([,T ],L # (R,L2 (T 2 f(p, (3.22 Z Zp L ([,T,L 2 (T 2 f(p, ( Reference solution. Having Fourier coefficients of Z on hand, we will do the same for function z (t, x solution to (. in order to compare it to the profile Z for a given, in a fixed time. The Fourier expansion of z is given by z (t, x, x 2 = m,n z m,n(t e 2πi(mx +nx 2, (4. where m =,, 2,... and n =,, 2,..., then the Fourier expansion of z is z = ż m,n(t e 2πi(mx +nx 2. (4.2 m,n Using the same idea as in the Fourier expansion of Z, we obtain the following infinite system of Ordinary Differential Equations z m,n (t ( 2iπA grad m i i,j (t z n j m i,n j (t i,j + 4π2 i,j A i,j (t((m i 2 + (n j 2 z m i,n j (t = Cm,n(t, (4.3 where A grad i,j (t, A i,j (t and C m,n(t are respectively the Fourier coefficients of A, A and C. In the same way, the truncated Fourier series of order P N of z is given by z P (t, x, x 2 = P m,n= which gives from (4.3 the following system Ordinary Differential Equations + 4π2 P z m,n (t i,j= P i,j= z m,n(t e 2πi(mx +nx 2, (4.4 ( 2iπA grad m i i,j (t n j z m i,n j (t A i,j (t((m i 2 + (n j 2 z m i,n j (t = Cm,n(t. (4.5 In (4.5, we will use an initial condition z m,n(, x. To solve (4.5 we use, for the discretization in time, a Runge-Kutta method (ode45.

7 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS Comparison Two-Scale Numerical Solution and reference solution. In this paragraph, we consider the truncated solution z P (t, x, x 2 and Z P (t, t, x, x 2. The objective here is to compare for a fixed and a given time, the quantity z P (t, x, x 2 Z P (t, t, x, x 2 when the water velocity U is given Comparisons of z P (t, x and Z P (t, t, x with U given by (4.6. For the numerical simulations, concerning z, we take z (x, x 2 = cos 2πx + cos 4πx and z (x, x 2 = Z(,, x, x 2. In what concerns the water velocity field, we consider the function U(t, θ, x, x 2 = sin πx sin 2πθ e, (4.6 where e and e 2 are respectively the first and the second vector of the canonical basis of R 2 and x, x 2 are the first and the second components of x. In Figure, we can see the space distribution of the first component of the velocity U for a given time t = and for various values of θ:.3,.55, and.7. In Figure 2, we see, for a fixed point x = (x, x 2, how the water velocity Ũ(θ hal-8732, version - 4 Oct Figure. Space distribution of the first component of U(,.3, (x, x 2, U(,.55, (x, x 2 and U(,.7, (x, x 2 when U is given by (4.6. evolves with respect to θ. In Figure 3, the θ-evolution of Ã(θ is also given in various points (x, x 2 R 2.

8 8 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK hal-8732, version - 4 Oct Figure 2. θ-evolution of Ũ(θ, (/2, and Ũ(θ, (/4, when U is given by ( , Figure 3. θ-evolution of Ã(θ, (/2, and Ã(θ, (/4, when U is given by (4.6 In this paragraph, we present numerical simulations in order to validate the Two-Scale convergence presented in Theorem.. For a given, we compare Z P (t, t, x, where Z P is the Fourier expansion of order P of the solution to (. and zp (t, x the Fourier expansion of order P of the solution to the reference problem. The simulations presented are given for P = 4. The calculation of zp (t, x implies knowledge of z (x. For an initial condition z (x well prepared and equal to Z(,, x, we obtain the results of Figure 4 and we remark that the results obtained are the same for zp (t, x and Z P (t, t, x.

9 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS hal-8732, version - 4 Oct 23 Figure 4. Comparison of zp (t, and Z P (t, t,, P = 4, at time t =, =., when U is given by (4.6 and when z ( = Z(,,. On the left zp (t,, on the right Z P (t, t,. In practice, the solution Z P, P N evolves according to P. For the simulations, we made the value of the integer P vary and we saw that this variation is very low from P 6. To better show that Z P (t, t, x, x 2 is close to the reference solution zp (t, x, x 2, we plot and compare Z P (t, t, x, and zp (t, x,, at different times t. In these comparisons the initial condition z (x, x 2 = cos 2πx + cos 4πx is different from Z(,, x, x 2. The results are shown in Figure 5 and Figure 6. We see in these figures that the solution zp (t, x get closer and closer to Z P (t, t, x with time of order. 4 x 3 Z(t,t/ε,x, and z ε (t,x,, t=ε, ε=. 4 x 3 Z(t,t/ε,x, and z ε (t,x,, t=2ε, ε=. 3 z ε 3 z ε 2 2 Z Z.3 Z(,,x and z (x.2 z (x Z(,,x Figure 5. Comparison of z P (t, x, and Z P (t, t, x,, P = 4. On the left t =, in the middle t = and t = 2 on the right, =..

10 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK 4 x 3 Z(t,t/ε,x, and z ε (t,x,, t=ε, ε=. 4 x 3 Z(t,t/ε,x, and z ε (t,x,, t=2ε, ε=. 3 z ε 3 z ε 2 2 Z Z.3 Z(,,x and z (x.2 z (x Z(,,x. hal-8732, version - 4 Oct Figure 6. Comparison of z P (t, x, and Z P (t, t, x,. On the left t =, in the middle t = and t = 2 on the right, =.. So we can see from these figures that the solution Z of the Two-Scale limit problem is such that Z(t, t,, is close to the solution z (t,, of the reference problem. In the presently considered case where the initial condition for z is not Z(,,,, we saw in Figure 5 and Figure 6 that zp tends to reach a steady state. This steady state is an oscillatory one in the sense that for large t, zp (t,, behaves like Z P (t, t,,. This is illustrated by Figure 7 where zp (t, x, and Z P (t, t, x, are given for various value of t in a period of lenght. More precisely, in this figure we see that within a period of time of lenght, zp (t,, and Z P (t, t,, do not glue together completly. Nevertheless, despite this phenomenon which is linked with the fact that the Two-Scale approximation of z (t,, by Z(t, t,, is only of order in, the two solutions re-glue well together at the end of the period.

11 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS t=.3.2 t=+ε/ hal-8732, version - 4 Oct t=+ε/ t= t=+3ε/ t=+ε/ t=+ε/4. t=+3ε/ Figure 7. Evolution of Z P (t, t, x, in the top and zp (t, x, in the bottom, t = + n 4, n =,, 2, 3.

12 2 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK Comparisons of z (t, x and Z(t, t, x with U is given by (4.7. In this subsection, we do the same as in the precedent one, but when the velocity fields U given by (4.7. The results are all identical to the precedent one i.e. the Two-Scale limit Z P (t, t, x, x 2 is very close to the solution zp (t, x, x 2 to the reference problem when P = 4. The initial condition z (x, x 2 Z(,, x, x 2 and is the same as in the subsection The results are given for =. and =.5 and for various time t. We notice that z comes very close to Z(t, t, x, x 2 when is very small. We begin by giving the space distribution of U at various time and the θ evolution of U and Ã. The second velocity fields is given by in [, θ ], θ θ θ 2 θ U thr e 2 in [θ, θ 2 ], U thr e 2 + φ( θ θ 2 θ 3 θ 2 ψ(t, x in [θ 2, θ 3 ], hal-8732, version - 4 Oct 23 U(t, θ, x, x 2 = U(t, θ, x = θ θ 3 θ 4 θ 3 U thr e 2 in [θ 3, θ 4 ], in [θ 4, θ 5 ], θ θ 5 θ 6 θ 5 U thr e 2 in [θ 5, θ 6 ], U thr e 2 φ( θ θ 6 θ 7 θ 6 ψ(t, x in [θ 6, θ 7 ], θ θ 7 θ 8 θ 7 U thr e 2 in [θ 7, θ 8 ], in [θ 8, ], where U thr >, φ is a regular positive function satisfying φ(s = s( s and ψ(t, x = ( + sin π 3 t(u thre 2 + ( + sin 2πx e, θ i = i+, i =,..., 8. The θ-evolution of U, given by (4.7, is given in Figure 9 for various position in [, ] 2. Function g a(u = g c(u = u 3, a = c = and M(t, θ, x = which yields a θ-evolution of Ã(θ which is drawn for various positions in Figure. (4.7

13 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS hal-8732, version - 4 Oct Figure 8. Space distribution of the first component of U(,.25, (x, x 2, U(,.275, (x, x 2 and U(,.75, (x, x 2 when U is given by (4.7. Using this, we compute Z P (t, t, x, x 2 and zp (t, x for P = 4. To compute z P (t, x we take z (x, x 2 = cos 2πx + cos 4πx which is not Z(,, x, x 2. First we study the errors Z P (t, t, x, x 2 zp (t, x at t =. This quantity decreases when decreases as illustrated in the following tabular. value of norm L norm L 2 norm L Table: Errors norm Z P (t, t, x, x 2 z P (t, x, x 2, P = (4, 4, P = (4, 4, 4, t =. The results given in this table show that, at time t =, z (t, x is closer to Z(t, t, x when is very small. These results validate the results obtained in Theorem.. In Figures and 2, we present simulations at times t =.75 and t =.775. We see that Z P (t, t, x, x 2 is close to zp (t, x, x 2. The numerical results shown in these figures are made with =

14 4 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK hal-8732, version - 4 Oct Figure 9. θ-evolution of U(, θ, (,, U(, θ, (4, and U(, θ, (/3, /3 when U is given by (4.7. In Figure 3 and 4, we do the same but for =.5. The numerical results show that zp (t, x is also very close to Z P (t, t, x, x 2.

15 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS hal-8732, version - 4 Oct Figure. θ-evolution of Ã(, θ, (,, Ã(, θ, (4, and Ã(, θ, (/3, /3 when U is given by (4.7. We remark that for =. and =.5, the solution zp (t, x is very close to Z P (t, t, x. But the approximation zp (t, x Z P (t, t, x is very good when is very small. To show that zp is very close to Z P, we construct the same figures as previously but in dimension 2 with =.5 i.e. we construct zp (t, x, and Z P (t, t, x, for =.5 at time t =.775. This is given in Figure 5.

16 6 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK hal-8732, version - 4 Oct 23 Figure. Comparison of z P (t, x, x 2 and Z P (t, t, x, x 2, P = 4; t =.75, =., z (x, x 2 = cos 2πx + cos 4πx. On the left Z P (t, t, x, x 2, on the right z P (t, x, x 2.

17 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS 7 hal-8732, version - 4 Oct 23 Figure 3. Comparison of z P (t, x, x 2 and Z P (t, t, x, x 2, P = 4; t =.75, =.5, z (x, x 2 = cos 2πx +cos 4πx. On the left Z P (t, t, x, x 2, on the right z P (t, x, x 2.

18 8 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK Z(t,t/ε,x,, ε=.5, t=.775 z ε (t,x,, ε=.5, t= hal-8732, version - 4 Oct Figure 5. Comparison of z P (t, x, and Z P (t, t, x,, t =.775, =.5. On the left Z P (t, t, x,, on the right z P (t, x,. The results in Figure 6 show that Z P and zp have the same behavior in the same period and Z P is very close to zp. We also notice that, despite the small shifts that occur during a period, the two solutions glue together.

19 TWO-SCALE NUMERICAL SIMULATION OF SAND TRANSPORT PROBLEMS 9 Z(t,t/ε,x,, t=,ε=.5.4 Z(t,t/ε,x,, t=+ε/4, ε= hal-8732, version - 4 Oct Z(t,t/ε,x,, t=+ε/2, ε= Z(t,t/ε,x,, t=+3ε/4, ε= z ε (t,x,, t=,ε=.5.4 z ε (t,x,, t=+ε/4, ε=.5

20 2 IBRAHIMA FAYE, EMMANUEL FRÉNOD AND DIARAF SECK REFERENCES hal-8732, version - 4 Oct 23 [] G. Allaire, Homogenization and Two-Scale convergence, SIAM J. Math. Anal. 23 (992, [2] P. Aillot, E. Frénod, V. Monbet, Long term object drift in the ocean with tide and wind. Multiscale Modelling and Simulation, 5, 2(26, [3] I. Faye, E. Frénod, D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment,discrete and Continuous Dynamical Systems, Vol 29 N o 3 March 2, pp -3. [4] E. Frénod, A. Mouton, Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates. Journal of Pure and Applied Mathematics: Advances and Applications, Vol 4, No 2 (2, pp [5] E. Frénod, A. Mouton, E. Sonnendrücker Two-Scale numerical simulation of the weakly compressible D isentropic Euler equations. Numerishe Mathematik,(27 Vol 8, No2, pp (DOI :.7/s [6] E. Frénod, F. Salvarani, E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a Two-Scale PIC-method. Mathematical Models and Methods in Applied Sciences, Vol. 9, No 2 (29 pp (DOI No:.42/S [7] E. Frénod, Raviart P. A., and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field, J. Math. Pures et Appl. 8 (2, [8] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural ceva, Linear and quasi-linear equations of parabolic type, AMS Translation of Mathematical Monographs 23 (968. [9] A. Mouton, Approximation multi-échelles de l équation de Vlasov, thèse de doctorat, Strasbourg 29. [] A. Mouton, Two-Scale semi-lagrangian simulation of a charged particules beam in a periodic focusing channel, Kinet. Relat. Models, 2-2(29, [] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 2 (989, address: ibrahima.faye@uadb.edu.sn address: emmanuel.frenod@univ-ubs.fr address: diaraf.seck@ucad.edu.sn