A new type of high-order finite difference compact reconstruction multi-resolution WENO scheme for nonlinear degenerate parabolic equations

In recent years, with the gradual improvement of the floating-point computing capabilities, the precise simulation of multi-scale flow structures such as turbulence has become a hot topic. The resolution and compactness of the scheme have become an important property. Thus, a compact reconstruction multi-resolution weighted essentially non-oscillatory (CRMR-WENO) scheme with increasingly higher order of accuracy is presented to solve nonlinear degenerate parabolic equations in this paper. The scheme yields superior resolution and lower truncation errors in comparison to the classical WENO scheme (Liu et al., SIAM J Sci Comput 33(2):939-965, 2011). By choosing unequal-sized hierarchical stencils, the scheme avoids the appearance of the negative linear weights and the application of the mapped nonlinear weights. When constructing the high-order scheme, we do not need to reselect the stencils, just add a large compact stencil. This makes the construction of higher order schemes simpler. A large number of benchmark numerical tests including the degenerate parabolic convection-diffusion equation and the porous medium equation are presented to show the advantages of this new CRMR-WENO scheme over the classical WENO scheme.