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

Olivier Bokanowski (1, 2) , Giorevinus Simarmata (1)

1 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
2 LJLL - Laboratoire Jacques-Louis Lions

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 elements. It follows the ideas of the recent works of Crouseilles, Mehrenberger and Vecil (2010) and of Qiu and Shu (2011), for first order equations, based on exact integration, quadrature rules, and splitting techniques. In particular we obtain high order schemes, unconditionally stable and convergent, in the case of linear second order PDEs with constant coefficients. In the case of non-constant coefficients, we construct "almost" unconditionally stable second order schemes and give precise convergence results. The schemes are tested on several academic examples, including the Black and Scholes PDE in finance.

Mots clés: Semi-Lagrangian scheme; discontinuous Galerkin elements; method of characteristics; Black and Scholes equation

Domaines: Analyse numérique [math.NA]

Olivier Bokanowski, Giorevinus Simarmata. Semi-Lagrangian discontinuous Galerkin schemes for some first and second order partial differential equations. 2012, pp.DOI 10.1007/s10915-012-9648-x. ⟨hal-00743042v1⟩