High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter

Abstract

A new approach to the derivation of local extremum diminishing finite element schemes is presented. The monotonicity of the Galerkin discretization is enforced by adding discrete diffusion so as to eliminate all negative off-diagonal matrix entries. The resulting low-order operator of upwind type acts as a preconditioner within an outer defect correction loop. A generalization of TVD concepts is employed to design solution-dependent antidiffusive fluxes which are inserted into the defect vector to preclude excessive smearing of solution profiles by numerical diffusion. Standard TVD limiters can be applied edge-by-edge using a special reconstruction of local three-point stencils. As a fully multidimensional alternative to this technique, a new limiting strategy is introduced. A node-oriented flux limiter is constructed so as to control the ratio of upstream and downstream edge contributions which are associated with the positive and negative off-diagonal coefficients of the high-order transport operator, respectively. The proposed algorithm can be readily incorporated into existing flow solvers as a ‘black-box’ postprocessing tool for the matrix assembly routine. Its performance is illustrated by a number of numerical examples for scalar convection problems and incompressible flows in two and three dimensions.