Semi-Lagrangian discontinuous Galerkin schemes for some first- and second-order partial differential equations
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 6, pp. 1699-1730.

Explicit, unconditionally stable, high-order schemes for the approximation of some first- and second-order linear, time-dependent partial differential equations (PDEs) are proposed. The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin (DG) elements. It follows the ideas of the recent works of Crouseilles et al. [N. Crouseilles, M. Mehrenberger and F. Vecil, In CEMRACS’10 research achievements: numerical modeling of fusion. ESAIM Proc. 32 (2011) 211–230], Rossmanith and Seal [J.A. Rossmanith and D.C. Seal, J. Comput. Phys. 230 (2011) 6203–6232], for first-order equations, based on exact integration, quadrature rules, and splitting techniques for the treatment of two-dimensional PDEs. For second-order PDEs the idea of the scheme is a blending between weak Taylor approximations and projection on a DG basis. New and sharp error estimates are obtained for the fully discrete schemes and for variable coefficients. In particular we obtain high-order schemes, unconditionally stable and convergent, in the case of linear first-order PDEs, or linear second-order PDEs with constant coefficients. In the case of non-constant coefficients, we construct, in some particular cases, “almost” unconditionally stable second-order schemes and give precise convergence results. The schemes are tested on several academic examples.

Received:
Accepted:
DOI: 10.1051/m2an/2016004
Classification: 65M12, 65M15, 65M25, 65M60
Mots-clés : Semi-Lagrangian scheme, weak Taylor scheme, discontinuous Galerkin elements, method of characteristics, high-order methods, advection diffusion equations
Bokanowski, Olivier 1, 2; Simarmata, Giorevinus 3

1 Laboratoire Jacques-Louis Lions, Université Paris-Diderot (Paris 7), 75205 Paris cedex 13, France.
2 Unité de Mathématiques Appliquées, ENSTA ParisTech, 91120 Palaiseau, France.
3 Finance RI Department – Rabobank International, Europalaan 44, 3526 KS, Utrecht, The Netherlands.
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Bokanowski, Olivier; Simarmata, Giorevinus. Semi-Lagrangian discontinuous Galerkin schemes for some first- and second-order partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 6, pp. 1699-1730. doi : 10.1051/m2an/2016004. http://www.numdam.org/articles/10.1051/m2an/2016004/

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