Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
Date
2018Metadata
Show full item recordAbstract
\noindent We develop a 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
$C([0,T];L_{loc}^p(\mathbb{R}^N))$ 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.