Finite Volume Discretization of a Class of Parabolic Equations with Discontinuous and Oscillating Coefficients

Transcription

1 Applied Mathematical Sciences, Vol, 8, no 3, - 35 HIKARI Ltd, wwwm-hikaricom Finite Volume Discretization of a Class of Parabolic Equations with Discontinuous and Oscillating Coefficients Bienvenu Ondami Université Marien Ngouabi Faculté des Sciences et Techniques BP 69, Brazzaville, Congo Copyright c 8 Bienvenu Ondami This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract In this paper we consider the numerical approximation of a class of parabolic equations with highly oscillating and discontinuous coefficients We use the method of lines with a finite volumes approach to discretize this problem This discretization leads to an ordinary differential equation (ODE We then analyze two families of numerical schemes corresponding to explicit and implicit discretization of the obtained ODE Numerical results comparing the approximation obtained by this method and the homogenized problem s solution are presented They show that this method is accurate and robust for the class of problems studied Mathematics Subject Classification: 65M8, 65M Keywords: Parabolic equation, Finite volumes method, Method of lines, Homogenization Introduction We consider a parabolic equation modeling a diffusion process in a periodic medium (for example, an heterogeneous domain obtained by mixing period-

2 Bienvenu Ondami ically two different phases, one being the matrix and the other the inclusions To fix ideas, the periodic domain is called Ω, a bounded open set in R n (n =,, 3, with a smooth boundary Γ Its period (a positive number wich is assumed to be very small in comparison with size of the domain, and the rescaled unit periodic cell Y = (, n For a final time T >, a source term s(x, t, and an initial condition f(x, we consider the Cauchy problem: (P ( x u ( ( x c t div k u = s(x, t in Ω, u = on Γ (, T, u (x, = f(x in Ω k is a symetric and uniformly positive definite matrix in Ω wich has jumps discontinuities across a given interface c is a positive and uniformly bounded function wich has jumps discontinuities across a given interface The case of piecewise constant coefficients matrix k and piecewise constant function c is very important for the applications Study of the problem can be done by different methods When is small enough, from homogenization tolls (see, eg [], [4], [5], [6], [7], the original problem (P can be replaced by the homogenized problem, modeling some average quantity without the oscillations Whenever homogenized equations are applicable they are very useful for computational purposes When is not small, the original equation have to be approximated directly The numerical approximation of partial differential equations with highly oscillating coefficients has been a problem of interest for many years and many methods have been developped The case of elliptic equations where the matrix k has continuous coefficients is the most studied (see, eeg [3], [5],[6] and the bibliographies therein The case of discontinuous coefficients has been addressed recently in [4] and a long before in some works as [3], [6], [7] and [6] In this paper we are going to addres (P by the method of lines by using a finite volumes approach For a more advanced presentation of the method of lines and finite volume methods, the reader is refferred to [] and [8] and to [8] and [9], respectively Our study is limited to one-dimensional problem (-D problem, and the obtained results can be generalized in the two-dimensional problem The paper is organized as follows In section, a description of used methods is presented Section 3 is devoted to numerical simulations Lastly, some concluding remarks are presented in section 4

3 Finite volume discretization of a class of parabolic equations 3 Methods Our study will focus on the -D problem and we assume (without loss of generality that Ω = ], [ In this case the problem (P is written simply Q T := { < x <, < t T }, c u t(x, t (k (x u x x = s(x, t in Q T, u (, t = u ( (, t =, t ], T ], u (x, = f(x, x ], [, ( x ( x where k (x = k = k(y, c (x = c = c(y, with y = x, k and c are discontinuous and periodic functions of period on ], [ In all of this paper we make the following assumptions: (A α k(y β, ae in ], [ ; α, β R +, (A s L (Q T, (A3 f L (], [, (A4 c < c(y c + <, ae in ], [, with some c, c + R + Under the assuptions (A (A4, it is a well-know result that there exists a unique solution u of ( Furthermore, from the homogenization theory, when tends to zero, u tends to u (x, t u(x, t the solution of the following homogenized problem c u t (x, t (k u x x = s(x, t in Q T, u(, t = u(, t =, t ], T ], ( u(x, = f(x, x ], [, where k is the mean harmonic value of k(y on Y = (, and c is the mean arithmetic value of c(y on Y = (, In order to compute a numerical approximation of u, we are going to beging by a discretization of the problem ( only in space, wich leads us to a ODE in time This approach is also called method of lines The obtained ODE will be analysed by two families of numerical schemes corresponding to explicit an implicit discretization Semi-discrete approximation By semi-discretization we mean discretization only in space, not in time We discretize (, into N equal size grid cells of size h = /N, and de-

4 4 Bienvenu Ondami fine ( x i = h/ + i h, i =,, N, so that x i is the center of cell I i = xi /, x i+/ with x i=,,n =, x N = Definition Let T = (I i i=,,n, where I i is given above T is called admissible uniform mesh of (, In a finite volume method the unknowns approximate the average of the solution over a grid cell More precisely, we let u i (t be the approximation u i (t := h xi+/ x i / c u (x, tdx; where u i (t h xi+/ Integrating ( over the cell I i and dividing par h we get h xi+/ x i / c (xu t(x, tdx = h xi+/ x i / (k (x u x x dx+ h x i / u (xdx, and := c (x i xi+/ x i / s(x, tdx (3 Let T = (I i i=,,n be an admissible uniform mesh of (, in the sens of Definition ( such that the discontinuities of k and c coincide with the interfaces of the mesh First integral in the right-hand term of (3 becomes: xi+/ x i / (k (x u x x dx = (k (x u x (x i+/, t (k (x u x (x i /, t (4 In order that the scheme be conservative, the discretization of the flux k (x u x at the point x i+ should have the same value on I i and I i+ To this purpose, one introduces as in [8], [9] and [4], the auxiliary unknown u (approximation of u at x i+ i+ Since on I i and I i+, k is continuous, the approximation of k u x may be perform on each side of x i+ by using the finite difference principle: F + (t = (k (x u x (x + k i F + (t = (k (x u x (x + k i+ u i+ (t u i (t h u i+(t u i+ (t h on I i, on I i+, where ki (repectively ki+ is the value of k (x ( on I i (repectively on I i+ Requiring the two above approximation of (k u x, t, one deduces: x i+ u + (t = k i+u i+ (t k i u i (t k i+ + k i (5

5 Finite volume discretization of a class of parabolic equations 5 So we get: where F (t = τ + i+ τ i+ u i+(t u i (t, (6 h = k i ki+ ki+ + (7 k i From (3, (4, (6 and (7 we get the following scheme for the inner points x i, i N, So du i (t dt du i (t dt = h ( F (t F (t + s + i (t, = [ ( τ h i+ u i+ (t u i (t ( τ i u i (t u i (t ] + s i (t du i dt = ( ] [τ u h i i τ + τ u i i+ i + τ u i+ i+ +s i, i =,, N (8 To complete the scheme (8 we need update formula also for the boundary points i = and i = N These must be derived by taking the boundary conditions into account We introduce the ghost cells I and I N wich located just outside the domain The boundary conditions are used to fill these cells with values u and u N, based on the values u i in the interior cells The same update formula (8 as before can then be used also for i = and i = N Let us consider our boundary conditions u (, t = and u (, t = Since the center of cells I and I N are not on the boundary, we take the average of two cells to approximate the value in between, and = u (, t u (x, t + u (x, t = u (, t u (x N, t + u (x N, t for i = and i = N, (8 becomes: c du N N dt c du dt = ( [τ u h = h [ τ N 3 u N from (9 and ( we get: τ + τ u (t + u (t u N (t + u N (t (9 ( ] u + τ u i+ + s, ( ( ] τ + τ u N 3 N N + τ u N N +s N ( u (t = u (t et u N(t = u N (t (3

6 6 Bienvenu Ondami from (3 one deduces: c du dt = [ ( h τ + τ ] u + τ u i+ + s (4 c du N N dt = [ ( ] τ u h N 3 N τ + τ u N 3 N N + s N (5 from (8, (4 and (5 we get the ODE du (t dt = A u (t + S(t, (6 where A is given by: A = (τ c h + τ, A = τ c h, A i i = τ i, i N, h τ i+ A i i+ =, i N, h A i i = (τ + τ, i N, h i i+ τ A N N = N 3 c, A N h N N = (τ + τ, c N h N 3 N (7 and S(t = s (t c s (t c s N (t c N s N (t c N (8

7 Finite volume discretization of a class of parabolic equations 7 Fully discrete approximation The system (6 can be solved by different methods In this paper we examine two numerical schemes corresponding to explicit and implicit discretization We note t = T M (M >, the time step Explicit scheme The forward Euler method to solve the ODE (6 is given by: Un+ = Un+ t [A Un + S n ], Un U (t n, S n S(t n, t n = n t, n =,, M, (9 where U (t = u (t u (t u N (t u N (t As for any ODE method, there arises the question of stability whose the answer is given by the following proposition Proposition Let C = i N ( τ i + τ i+ and T = ( I i i=,,n be an admissible uniform mesh of (, in the sens of Definition, such as the discontinuities of k and c coincide with the interfaces of the mesh Then under the CFL condition: CF L := t h C, the explicit scheme (9, where the matrix A is given by (7 and S(t is given by (8 is L stable Proof The proof is identical to that given in [3] We write the scheme (9 in the following form for the inner points: u n+, i = t [ c τ u i h i n, i + t (τ ] + τ u h i i+ n, i+ t c τ u i h i+ n, i++ t s n, i i N (

8 8 Bienvenu Ondami The key point of the proof is that, under the CF L condition, the right-hand side of the equation ( is a convex combinaton of u n, i, u n, i and u n, i+, plus the term t s n i From ( on can deduce: u n+, i t c τ i h i u n, i [ + t (τ ] + τ u h i i+ n, i + t c τ i h i+ u n, i+ i N Using (, ( and (3 we get the previous inequaly for i = and i = N One deduces [ u n+, i t c τ + t (τ + τ + t ] i h i h i i+ c τ u i h i+ n + t c sn i So U n+ By obvious recurrence, we get U n U +n t c Hence the L stability is proven Un + t c S k k M S k U + T k M c + t S k, n {,, M} k M Remark Proposition gives a time step restriction of the type t Ch This is a severe restriction, which is seldom warranted from an accuracy point of view It leads to unnecessarily expensive methods Implicit scheme The implicit Euler scheme to solve the system (6 is given by: U n+ = U n+ t [ A U n+ + S n+], U n U (t n, S n S(t n, t n = n t, n =,, M ( So (I ta U n+ = U n + ts n+, so U n+ = (I ta [ U n + ts n+] ( Proposition 3 Let T = ( I i i=,,n be an admissible mesh of (, in the sens of Definition, such as the discontinuities of k and c coincide with the interfaces of the mesh Then the implicit scheme (, where the matrix A is given by (7 is L stable s n i,

9 Finite volume discretization of a class of parabolic equations 9 Proof We write the scheme ( in the following form for the inner popints: [ + t (τ ] + τ u h i i+ n+, i t c τ u i h i n+, i t c τ u i h i+ n+, i+ = u n, i+ t s n+, i We are going to prove that: and i N u n, i min i N u n, i Let i such as u n+, i = i N (3 N i N u, i + t p= N min i N u, i + t i N u n+, i p= i N min i N Then u n+, i u n+, j, for j = i ± From (3, we have In particular i N u n+, i By recurrence and from (6 we get i N u n, i u n+, i u n, i + t s n+ i i N u n, i + t N i N u, i + t p= i N i N ( s p i, (4 ( s p i (5 ( s n+ i (6 ( s p i (7 Using (, (, (3 and (7, we get (4 The proof of (5 can be done by simular way Hence the L stability of the scheme ( is proven 3 Numerical simulations In this section, we are going to present numerical results obtained by using the implicit scheme (, and compare them with the homogenized solution (Homog More especially, we shall present numerical results obtained with following data: k (y = { k if < y < k if < y <, k (x = k, if p < x < ( p + ( k, k R + k, if p + < x < (p +,

10 3 Bienvenu Ondami { c if < y < c (y = c if < y <, c (x = = n p, where n p is a positive integer, < p < n p c, if p < x < ( p + c, if ( c, c R + p + < x < (p +, First test problem involved simulations with k = 4, k =, c =, c = 5, the source function is s(x, t =, and the initial condition function is f(x = The below graphics of Figure confirm that when is enough small the homogenized solution gives a good approximation of the original problem s solution, etherwise a direct approximation of the original problem is required NS Homog 6 NS Homog Figure : Test problem Graphics of Figure give the temperature distribution after 5, 35 and 45 seconds (graphic on left and the temperature distribution for = 8, = 6 and = 3

11 Finite volume discretization of a class of parabolic equations NS: T=5 s NS: T=35 s NS: T=45 s Figure : Test problem In the below second test problem, we changed the specific heat: c = 6 and c = 48; t = All of the others data remain the same as in the first test problem The left one graphic gives the temperature distribution after and 3 seconds and the right one gives the temperature distribution after 8 and seconds and shows that the numerical solution is stable from 8 seconds

12 3 Bienvenu Ondami Figure 3: Test problem The last series of tests was done with following data: k =, k = 3, c =, c = 4, the initial condition function f(x =, and the source function { if < y < s (x, t =, if < y < The below left one graphic gives the temperature distribution after 5, 5, 8 and seconds and shows again the stabilty of the numerical solution from 8 seconds, and the right one graphic gives the temperature distribution for = 5, = 5 and =

13 Finite volume discretization of a class of parabolic equations Figure 4: Test problem 3 4 Concluding remarks The purpose of this paper was to apply the method of lines with a finite volumes approach for a class of parabolic equations with discontinuous and oscillating coefficients We studied and analyzed numerical explicit and implicit schemes, for solving the ODE obtained After proving that the explicit scheme is L stable under CFL condition, and that the implicit scheme is L stable without any CFL condition, we presented numerical results obtained by using the implicit scheme, showing that this approach is well adapet to the discretization of this class of problems The extension of this technique to the two-dimensional problem with uinform rectangular grids is straightforward Acknowledgements The author would like to thank the editor and anonymous reviewer for their positive comments References [] A Absule and E Weinan, Finite difference heterogeneous multi-scale method for homogenization problems, Journal of Computional Physics, 9 (3,

14 34 Bienvenu Ondami [] G Allaire, Introduction to Homogenization Theory, CEA-EDF-INRIA School on Homogenization, 3-6 December allaire/homog/lectpdf [3] B Amaziane and B Ondami, Numerical approximations of a second order elliptic equation with highly oscillating coefficients, Notes on Numerical Fluid Mechanics, 7 (999, - [4] N Bakhvalov and G Panasenko, Homogenization: Averaging in Periodic Media, Vol 36, Springer, Dordrecht, [5] A Bensoussan, JL Lions and G Papanicolaous, Asymptotic Analysis for Periodic Structures, Vol 5, North-Holland, Amsterdan, 987 [6] JF Bourgat, Numerical Experiments of the Homogenization Method for Operators with Periodic Coefficients, INRIA, Rapport de Recherche N o 7, 978 [7] JF Bourgat and A Dervieux, Méthode D homogénéisation des Opérateurs à Coefficients Périodiques: Étude des Correcteurs Provenant du Développement Asymptotique, INRIA, Rapport de Recherche N o 78, 978 [8] R Eymard, T Gallout and R Herbin, The finite volume methods, Chapter in Techniques of Scientific Computing, Handbook of Numerical Analysis, PG Ciarlet and JL Lions, editors, Vol VII, III,, [9] R Eymard, T Gallout and R Herbin, The Finite Volume methods, Update of the preprint wwwcmifr [] S Hamdi, WE Schiesser and GH Griffiths, Method of lines, Scolarpedia, (7, no 7, [] U Hornung, Homogenization and Porous Media, Vol 6, Springer-Verlag, New York, [] VV Jikov, SM Kozlov and OA Oleinik, Homogenization of Differentials Operators and Integral Functionals, Springer-Verlag, Berlin, [3] B Ondami, The Method of Lines with a Finite Volumes Approach for Transient Convection-Diffusion Problems, Journal of Mathematical Sciences: Advances and Applications, 44 (7,

15 Finite volume discretization of a class of parabolic equations 35 [4] B Ondami, Approximation of a Second-order Elliptic Equation with Discontinuous and Highly Oscillating Coefficients by Finite Volume Method, Journal of Mathematics Research, 8 (6, no 6, [5] B Ondami, The Effect of Numerical Integration on the Finite Element Approximation of a Second Order Elliptic Equation with Highly Oscillating Coefficients, Journal of Interpolation and Approximation in Scientific Computing, 5 (5, [6] B Ondami, Sur Quelques Problèmes D homogénéisation des Écoulements en Milieux Poreux, Ph D Thesis, Université de Pau et des Pays de l Adour, [7] R Orive and E Zuazua, Finite difference approximation of homogenization problems for elliptic equations, Multiscale Model and Simul, 4 (5, no, [8] O Runborg, Finite Volume Discretization of the Heat Equation, KTH Computer Science and Communication [9] E Sanchez-Palancia, Non-Homogeneous Media and Vibration Theory, Vol 7, Springer-Verlag, Berlin, 98 [] H Versieux and M Sarkis, Convergence analysis for the numerical boundary corrector for elliptic equations with rapidly oscillating coefficients, SIAM J Numer Anal, 46 (8, no, [] H Wen-ming and C Jun-Zhi, A new approximate method for second order elliptic problems with rapdly oscillating coefficients based on the method of multiscale asymptotic expansions, J Math Anal and Applications, 335 (7, Received: July 4, 7; Published: February 5, 8