Adaptive high-order shock-capturing weno-θ finite difference method for nonlinear parabolic problems

Abstract

The sixth-order shock-capturing WENO-theta finite difference scheme developed for hyperbolic conservation laws is numerically investigated for nonlinear degenerate parabolic problems whose solutions may contain shocks or discontinuities. In addition to the nonlinear switching mechanism of the sub-stencils based on their smoothness measurement, the WENO-theta scheme further introduces a new parameter theta adaptively choosing the better choice among the fifth-order upwind and sixth-order central large stencils, which in turn improves the overall accuracy to sixth order in smooth regions whereas maintains sharp capturing of shocks or discontinuous regions. The parabolic term is numerically approximated using a direct conservative flux difference. Numerical results are given for a variety of problems including the well-known Burgers equation serving as a prototype of the Navier-Stokes equations in fluid dynamics, and the Buckley-Leverett equation which has a non-convex flux function and is the model equation of two-phase flows for oil reservoir simulation.