Multi-dimensional asymptotically stable finite difference schemes for the advection-diffusion equation

摘要

An algorithm is presented which solves the multi-dimensional advection-diffusion equation on complex shapes to 2nd-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by tradition definitions, fail. It gives accurate, non oscillatory results even when boundary layers are not resolved.