Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments

TL;DR

This paper develops and tests robust, convergent numerical schemes for nonlinear degenerate diffusion equations, including local and nonlocal types, with a focus on stability, convergence, and applicability to various nonlinearities and fractional operators.

Contribution

It introduces a unified framework for designing and analyzing numerical schemes for nonlocal and local porous medium type equations, including new high-order methods and discretizations for complex nonlinearities.

Findings

Numerical schemes are proven to be stable and convergent under weak assumptions.

New high-order and discrete Laplacian methods are introduced for nonlocal equations.

Numerical experiments confirm the effectiveness of the schemes on various problems.

Abstract

We develop unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}$ is a general symmetric L\'evy type diffusion operator. Included are both local and nonlocal problems with e.g. $\mathfrak{L}=\Delta$ or $\mathfrak{L}=-(-\Delta)^{\frac\alpha2}$, $\alpha\in(0,2)$, and porous medium, fast diffusion, and Stefan type nonlinearities $\varphi$. By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are $L^p$-stable for $p\in[1,\infty]$, compact, and convergent in $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$ for $p\in[1,\infty)$. The first part of this project is given in \cite{DTEnJa18a} and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of \cite{DTEnJa18a} apply and testing the schemes numerically. Our examples include fractional diffusions of different orders, and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems.