Modélisation Mathématique Et Analyse Numérique

In this paper we introducé the Local Projection P° P-Discontinuous Galerkin finite elemente method (ÂI1P ° P -scheme) for solving numencally scalar conservation laws This is an exphcit method obtained by modifying the expltcit Discontinuous Galerkin method introduced by G Chavent and G Salzano [3], via a simple local projection based on the monotomcity-preserving projections introduced by van Leer [13] The resulting scheme is an extension o f Godunov scheme that vérifies a local maximum pnnciple, and is TV DM (total variation diminishing in the means) Convergence to a weak solution is proven We display numencal évidence that the scheme is an entropy scheme of order one even when discontinuities are present Resumé —Nous proposons une méthode d'éléments finis discontinus P° P avec projection locale pour le calcul des lois de conservation scalaires C'est un schéma explicite obtenu en modifiant la méthode de Galerkin discontinue explicite, introduite par G Chavent et G Salzano [3], a l'aide d'une simple projection locale basée sur les projections introduites par Van Leer [13] qui garde ses propriétés de conservation de la monotomcité Le schéma correspondant est une extension du schéma de Godunov qui vérifie un principe du maximum localy et est DVTM (diminue la variation totale sur les moyennes) Nous démontrons la convergence vers une solution faible, et fournissons des résultats numériques montrant que le schema est entropique d'ordre un même en présence de discontinuité 1. INRODUCTION In this paper we introducé and analyze a new finite element method, the local projection P° P^Discontinuous Galerkin method (AILP°P^scheme), devised to solve numerically the scalar conservation law 3,w + dj(«) = 0 , on (0, r ) x R , u(t = 0) = u0, inR, K ' } where the nonhnear function/ is assumed to be C, and the initial data w0 is assumed to belong to the space L^R) n BV (R). This finite element (*) Received in December 1987 (0 INRIA, Domaine de Voluceau, Rocquencourt, B.P 105, 78153 Le Chesnay Cedex, France , and CEREMADE, Université Faris-Dauphme, 75775 Pans Cedex 16, France () IMA, University of Minnesota, 514 Vincent Hall, Minneapolis, Minnesota 55455, USA M AN Modélisation mathématique et Analyse numérique 0764-583X/89/04/565/28/$ 4 10 Mathematical Modellmg and Numencal Analysis © AFCET Gauthier-Villars 566 G CHAVENT, B COCKBURN method is a predictor-corrector method whose prédiction is given by the explicit P°F-Discontinuous-Galerkin method introduced by G Chavent and G Salzano in [3], and whose correction is obtained by means of a very simple local projection, that we shall call AH, based on the monotonicitypreservmg projection introduced by Van Leer in [13] The basic idea of this method is to write the approximate solution uh as the sum of a piecewise-constant function Uh, and a function uh whose restriction to each element has zero-mean, and to consider the method as a finite différence scheme for the means Uh The function üh is considered as a parameter The local projection All acts on the parameter üh, and is constructed in order to preserve the conservativity, and enforce the stabüity of the scheme for the means Uh In the extreme case m which the parameter üh is set identically equal to zero by the local projection Au, our scheme reduces to the well known Godunov scheme In the gênerai case, the scheme for the means keeps the local maximum pnnciple venfied by Godunov scheme, and is TVD (total variation dimimshing) Thus, the AHP ° P ̂ scheme is conservative, positive, and TVDM, i e total variation dimmishmg in the means We show that these properties, together with some properties of the local projection Au, imply the existence of a subsequence converging to a weak solution of (1 1) Our numencal results mdicate that if the cfl-numbei is mildly small enough, the scheme converges to the entropy solution with a rate of convergence equal to 1 m the L(0, T, L^J-norm even in the présence of discontinuities In 74 Le Samt and Ra^iart [9] introduced the Discontinuous-Galerkin method for solving the neutron transport équation |x dtu -f v dxu + cru = g They choose their approximate function to be piecewise a polynomial of at most degree k >: 0 in each of the variables t, and x In this way they obtained an ïmphcit scheme, but they did not had to solve ït globally Indeed, they proved that ït is possible to solve ït locally due to the fact that the direction of the propagation of the information, (|x, v), is always the same In the gênerai case, this is no longer true, for the local direction of propagation, (1, f'(u)), dépends on values that have not been calculated yet ' To overcome this difficulty, m 1978 G Chavent and G Salzano [3] modified this method and obtamed an explicit scheme that we shall call the P°P-Discontinuous-Galerkin method In this method the tand x-directions are treated in a different way the approximate solution is taken to be piecewise constant in time, and piecewise linear in space The two main advantages of the method are that ït is explicit, and that ït is very easy to generahze to the case of several space dimensions However, the scheme has a very restrictive stabüity condition — as we shall prove later —, and ït Modélisation mathématique et Analyse numérique Mathematical Modelhng and Numencal Analysis P°P-DISCONTINUOUS-GALERKIN FINITE ELEMENT 567 may not converge to the entropy solution in the case in which the nonlinearity ƒ is nonconvex — as the numerical évidence we shall display indicates. In 1984 one of the authors [4] modified the scheme and obtained a scheme called the G-l/2 scheme, for which the convergence to the entropy solution was proven in the gênerai case. A further development of the ideas involved in the construction of this scheme lead to the theory of quasimonotone schemes for which L°°(0, T ; L(R))-error estimâtes have been obtained ; see [5]. The scheme we now introducé can be considered as a simplification of the initial G-l/2 scheme. This simplification leads to a very simple, and much cheaper algorithm, but complicates enormously the proof of its convergence. At each time step the AILP ° P ̂ scheme consists of two phases : in the first, a prédiction is obtained by using the unchanged P°P ^method ; in the second, a correction is obtained by applying the local projection All to it. This projection dépends on a parameter, 0 e [0, 1], (0 may vary from element to element, but we have performed our numerical experiments with G = constant) and is based on the monotonicity-preserving local projections introduced by Van Leer in [13] : for 0 == 1 the All projection coincides with the one defined in [13, (66)] (thus, the AILP ° P ̂ scheme can be considered as a Discontinuous-Galerkin finite element version of the schemes introduced in [13]). One of the main contributions of this work is that we have proved that in fact the use of the local projection All — originally devised in order to produce positive and monotonicity-preserving schemes — renders the scheme under considération a TVDM scheme whose approximate solution vérifies a local maximum principle ; see Proposition 3.2. These two properties allow us to conclude that the scheme is indeed total variation bounded (TVB) and that it générâtes a subsequence converging in L°°(0, T ; L1 1 0C(IR)) to a weak solution of (1.1) ; see Theorem 3.3. The problem of pro ving that the weak solution is indeed the entropy solution is still open. A resuit in this direction is the proof of the convergence of MUSCL-type semidiscrete schemes in the case of a convex (or concave) nonlinearity by Osher in [10]. Also, Johnson and Pitkaranta [7] have analized the Discontinuous-Galerkin method in the linear case. An outline of the paper follows. In Section 2 we define the P° PDiscontinuous-Galerkin method, we obtain the L c/Z-stabüity condition for the linear case, and display some numerical expériences that show the typical behavior of the method. In Section 3 we define the local-projection P° P ^Discontinuous-Galerkin method, we obtain some stability properties, prove the convergence to a weak solution, and test it in the same examples the P° P ̂ Discontinuous-Galerkin method was tested. We end with some concluding remarks in Section 4. In what follows, the P° P ^DiscontinuousGalerkin method will be referred to simply by the P° P ^scheme, and the local-projection P° P ^Discontinuous-Galerkin by the AILPP ^scheme.