Semi-Lagrangian discontinuous Galerkin schemes for some first- and second-order partial differential equations
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.
Revised: 2 October 2015
Accepted: 15 January 2016
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.
Mathematics Subject Classification: 65M12 / 65M15 / 65M25 / 65M60
Key words: Semi-Lagrangian scheme / weak Taylor scheme / discontinuous Galerkin elements / method of characteristics / high-order methods / advection diffusion equations
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