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SEMILAGRANGIAN DISCONTINUOUS GALERKIN SCHEMES FOR SOME FIRST AND SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
, 2012
"... Abstract. Explicit, unconditionally stable, high order schemes for the approximation of some first and second order linear, timedependent partial differential equations (PDEs) are proposed. The schemes are based on a weak formulation of a semiLagrangian scheme using discontinuous Galerkin elements ..."
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Abstract. Explicit, unconditionally stable, high order schemes for the approximation of some first and second order linear, timedependent partial differential equations (PDEs) are proposed. The schemes are based on a weak formulation of a semiLagrangian 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 nonconstant 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. 1.