GODUNOV-MIXED METHODS FOR ADVECTIVE FLOW PROBLEMS IN ONE SPACE DIMENSION

A time-splitting method for solving advection-dominated, parabolic, partial differential equations is presented. In this method, a higher-order Godunov procedure approximates advection and a mixed finite element procedure approximates diffusion. Several variations on the basic scheme are formulated for solving one-dimensional, quasilinear, parabolic problems with Dirichlet boundary conditions. A maximum principle for one variant of the scheme is demonstrated, and L infinity (L2) and L2(L2) error estimates for the approximate solution and the diffusive flux, respectively, are derived. These estimates indicate that one variant of the scheme is L infinity-stable in certain situations, but possibly suboptimal in error, while another variant is optimal and L2-stable.