A New Type of High-Order Mapped Unequal-Sized WENO Scheme for Nonlinear Degenerate Parabolic Equations

In this paper, we propose the MUSWENO scheme, a novel mapped weighted essentially
non-oscillatory (WENO) method that employs unequal-sized stencils, for solving nonlinear
degenerate parabolic equations. The new mapping function and nonlinear weights are
proposed to reduce the difference between the linear weights and nonlinear weights.
Smaller numerical errors and fifth-order accuracy are obtained. Compared with traditional
WENO schemes, this new scheme offers the advantage that linear weights can be any
positive numbers on the condition that their summation is one, eliminating the need to
handle cases with negative linear weights. Another advantage is that we can reconstruct a
polynomial over the large stencil, while many classical high-order WENO reconstructions
only reconstruct the values at the boundary points or discrete quadrature points. Extensive
examples have verified the good representations of this scheme.