Semi-Lagrangian discontinuous Galerkin schemes for some first and second-order partial differential equations

Olivier Bokanowski (1) , Giorevinus Simarmata (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 Rabobank International - Finance RI Department

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, Mehrenberger and Vecil (2010), Rossmanith and Seal (2011), 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.

Mots clés: weak Taylor schemes; method of characteristics; advection diffusion equations; semi-Lagrangian scheme; discontinuous Galerkin elements; Black and Scholes equation; high order methods

Domaines: Analyse numérique [math.NA]

Olivier Bokanowski, Giorevinus Simarmata. Semi-Lagrangian discontinuous Galerkin schemes for some first and second-order partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis, 2016, 50 (6), 1699-1730 (32 p.). ⟨10.1051/m2an/2016004⟩. ⟨hal-00743042v4⟩