Finite volume WENO schemes for nonlinear parabolic problems with degenerate diffusion on non-uniform meshes

Abstract

We consider numerical approximation of the degenerate advection-diffusion equation, which is formally parabolic but may exhibit hyperbolic behavior. We develop both explicit and implicit finite volume weighted essentially non-oscillatory (WENO) schemes in multiple space dimensions on non-uniform computational meshes. The diffusion degeneracy is reformulated through the use of the Kirchhoff transformation. Space is discretized using WENO reconstructions with adaptive order (WENO-AO), which have several advantages, including the avoidance of negative linear weights and the ability to handle irregular computational meshes. A special two-stage WENO reconstruction procedure is developed to handle degenerate diffusion. Element averages of the solution are first reconstructed to give point values of the solution, and these point values are in turn used to reconstruct the Kirchhoff transform variable of the diffusive flux. Time is discretized using the method of lines and a Runge-Kutta time integrator. We use Strong Stability Preserving (SSP) Runge-Kutta methods for the explicit schemes, which have a severe parabolically scaled time step restriction to maintain stability. We also develop implicit Runge-Kutta methods. SSP methods are only conditionally stable, so we discuss the use of L-stable Runge-Kutta methods. We present in detail schemes that are third order in both space and time in one and two space dimensions using non-uniform meshes of intervals or quadrilaterals. Efficient implementation is described for computational meshes that are logically rectangular. Through a von Neumann (or Fourier mode) stability analysis, we show that smooth solutions to the linear problem are unconditionally L-stable on uniform computational meshes when using an implicit Radau IIA Runge-Kutta method. Computational results show the ability of the schemes to accurately approximate challenging test problems.