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: high order methods; Black and Scholes equation; weak Taylor schemes; method of characteristics; discontinuous Galerkin elements; Semi-Lagrangian scheme

Domaines: Analyse numérique [math.NA]

Olivier Bokanowski, Giorevinus Simarmata. Semi-Lagrangian discontinuous Galerkin schemes for some first and second order partial differential equations. [Research Report] Université Paris Diderot Paris 7. 2015. ⟨hal-00743042v2⟩