Boundary Mesh Refinement For Semi-Lagrangian Schemes

Abstract. We study semi-Lagrangian schemes for the Dirichlet problem for second-order degenerate elliptic PDEs. Like other wide stencil schemes, these schemes have to be truncated near the boundaries to avoid “overstepping”. The various modifications proposed in the literature lead to either reduced consistency orders for those points, or even a loss of consistency with the differential operator in the usual sense. We propose a local mesh refinement strategy near domain boundaries which achieves a uniform order of consistency up to the boundary in the first case, and in both cases reduces the width of the region where overstepping occurs, so that the practically observed convergence order is unaffected by overstepping. We demonstrate this numerically for a linear parabolic equation and a second order HJB equation.