A New Sixth-Order Finite Difference Compact Reconstruction Unequal-Sized WENO Scheme for Nonlinear Degenerate Parabolic Equations

Abstract

In this paper, a new sixth-order finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory (CRUS-WENO) scheme is designed for solving the nonlinear degenerate parabolic equations on structured meshes. This new CRUS-WENO scheme only uses the information defined on three unequal-sized spatial stencils, obtains the optimal sixth-order accuracy in smooth regions, and preserves the second-order accuracy near strong discontinuities. This scheme can be applied to dispose the emergence of the negative linear weights and avoid the application of the mapped function. The corresponding linear weights can be artificially set to be any random positive numbers as well as their summation is one. The construction process of this scheme is very simple and can be easily extended to higher dimensions, since the new compact conservative formulation is applied to approximate the second-order derivatives. The new CRUS-WENO scheme uses narrower large stencil than that of the same order finite difference classical WENO schemes. Therefore, it has a better compactness and smaller truncation errors. Some benchmark numerical tests including the porous medium equation and the degenerate parabolic convection-diffusion equation are performed to illustrate the advantages of this new CRUS-WENO scheme.