Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

TL;DR

This paper introduces a fully adaptive multiresolution finite volume scheme for one-dimensional strongly degenerate parabolic equations, enhancing computational efficiency while ensuring convergence to entropy solutions.

Contribution

It develops a novel adaptive multiresolution approach using a graded tree structure and optimal thresholding, improving efficiency for degenerate parabolic equations.

Findings

The scheme converges to the entropy solution.

Significant reduction in CPU time and memory usage.

Numerical tests confirm the method's efficiency and accuracy.

Abstract

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the $L^1$ contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties.