Extended generalized Lagrangian multipliers for magnetohydrodynamics using adaptive multiresolution methods

ESAIM: PROCEEDINGS, December 2013, Vol. 43, p. 95-107 S. Descombes, B. Dussoubs, S. Faure, L. Gouarin, V. Louvet, M. Massot, V. Miele, Editors EXTENDED GENERALIZED LAGRANGIAN MULTIPLIERS FOR MAGNETOHYDRODYNAMICS USING ADAPTIVE MULTIRESOLUTION METHODS M. O. Domingues 1 , A. K. F. Gomes 2 , S. M. Gomes 3 , O. Mendes 4 , B. Di Pierro 5 et K. Schneider 6 Résumé. Nous présentons une nouvelle méthode de multirésolution adaptative pour la simulation numérique de la magnétohydrodynamique idéale. Les équations qui régissent la dynamique, i.e., les équations d’Euler compressible couplées aux équations de Maxwell sont discrétisées suivant un schéma de type volumes finis sur un maillage cartésien en deux dimensions. L’adaptativité en espace est obtenue en utilisant une analyse de multirésolution en moyenne de cellule proposée par Harten, qui est une méthode fiable pour le raffinement local du maillage tout en contrôlant l’érreur. La discrétisation temporelle est un schéma de Runge-Kutta qui intègre un contrôle automatique du pas de temps. Pour imposer l’incompressibilié du champs magnétique, une approche par multiplicateur de Lagrange généralisé est utilisée, ici de type parabolique-hyperbolique. Pour illustrer les capacitées de cette méthode, des applications à des problèmes de Riemann ont été réalisées. Les coûts en mémoire sont présentés, et la précision de la méthode est évaluée par comparaison avec les solutions exactes du problème. Abstract. We present a new adaptive multiresoltion method for the numerical simulation of ideal magnetohydrodynamics. The governing equations, i.e., the compressible Euler equations coupled with the Maxwell equations are discretized using a finite volume scheme on a two-dimensional Cartesian mesh. Adaptivity in space is obtained via Harten’s cell average multiresolution analysis, which allows the reliable introduction of a locally refined mesh while controlling the error. The explicit time discretization uses a compact Runge–Kutta method for local time stepping and an embedded Runge-Kutta scheme for automatic time step control. An extended generalized Lagrangian multiplier approach with the mixed hyperbolic-parabolic correction type is used to control the incompressibility of the magnetic field. Applications to a two-dimensional problem illustrate the properties of the method. Memory savings and numerical divergences of magnetic field are reported and the accuracy of the adaptive computations is assessed by comparing with the available exact solution. 1 Laboratório Associado de Computação e Matemática Aplicada (LAC), Coordenadoria dos Laboratórios Associados (CTE), Instituto Nacional de Pesquisas Espaciais (INPE), Av. dos Astronautas 1758, 12227-010 São José dos Campos, São Paulo, Brazil (e-mail : , ) 2 Pós Graduação em Computação Aplicada, CTE, INPE (e-mail : ) 3 Universidade Estadual de Campinas (Unicamp), IMECC, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, 13083-859 Campinas, São Paulo, Brazil (e-mail : ) 4 Divisão de Geofı́sica Espacial(DGE), Coordenação de Ciências Espaciais(CEA), INPE (e-mail : ) 5 IRPHE–CNRS, Aix–Marseille Université, 49 rue F. Joliot-Curie, 13453 Marseille Cedex 13, France (e-mail : ) 6 M2P2–CNRS & Centre de Mathématiques et d’Informatique (CMI), Aix-Marseille Université, 38 rue F. Joliot–Curie, 13451 Marseille Cedex 20, France (e-mail : ) c EDP Sciences, SMAI 2013 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201343006 96 ESAIM: PROCEEDINGS Introduction Plasmas and electrically conducting fluids, ubiquitous in our daily life, are of major importance, as for example, in the Sun, which strongly influences the magnetic field around Earth, or for the dynamo action inside the liquid metal of Earth. The numerical simulation of magnetohydrodynamics (MHD) modeling such phenomena encounters, in comparison to ordinary hydrodynamics, additional complexities which are not only due to the presence of multiple spatial and temporal scales. The nonlocal character of boundary conditions of the magnetic field requires specific approaches, like matching the magnetic field at the boundary with the field in a surrounding vacuum, and in particular, the incompressibility of the magnetic field necessitates precise and efficient numerical techniques. Divergence errors in the magnetic field modify the underlying physics of the problem. Traditionally, Helmholtz decompositions, also called Hodge decompositions, are used to project the magnetic field onto divergence free fields. However, this approach is computationally expensive as it requires the solution of a Poisson equation, which implies the solution of very large linear systems especially in 3D. For a discussion on this topic we refer the reader e.g. to [2, 3, 20, 23]. Divergence cleaning techniques based on Lagrange multipliers have been introduced in the finite element context by Assous et al. [1] for time-dependent Maxwell equations. Since then different developments leading to various types of approaches can be found in the literature [2, 3, 19, 23]. Munz et al. [18] introduced a generalized Lagrangian multiplier formulation of Maxwell equations, which leads to different PDEs for the multiplier, being either of hyperbolic, parabolic, elliptic or mixed types. In the current work we use a variant of the previously cited approaches, the Extended Generalized Lagrange Multiplier (EGLM) designed by Dedner et al. [5] with a mixed parabolic-hiperbolic correction. The idea of the latter technique is not to enforce the divergence free condition exactly, but to promote a natural evolution of the system toward a divergence free state as discussed in [15]. The adaptive method of the present paper falls into the multiresolution (MR) category, which is designed to speed up finite volume schemes for time dependent conservation laws, based on ideas originally introduced in the work of Harten [13, 14]. The MR method is also combined with time adaptive strategies using either local or controlled time stepping. A review of multiresolution techniques can be found in the book of Müller [17], or in the review [10] and references therein. The aim of this paper is to combine for the first time the divergence cleaning technique for the magnetic field introduced by Dedner et al. [5] with adaptive multiresolution computations and to check its feasibility interplaying with adaptivity. The adaptive multiresolution code originally developed by Roussel et al. [21] has been extended to include Maxwell equations governing the magnetic field [12]. For divergence cleaning the EGLM formulation is used. The resulting new method is applied to a Riemann test problem for which the exact solution is known and the divergence of the magnetic field remains zero. The accuracy of the adaptive computations is assessed and their efficiency in terms of memory (...truncated)