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Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
- Source :
- BIRD: BCAM's Institutional Repository Data, instname, SIAM Journal on Numerical Analysis
- Publication Year :
- 2018
-
Abstract
- \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.<br />Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway - ERCIM “Alain Bensoussan” Fellowship programme - “Juan de la Cierva - formación” program (FJCI-2016-30148)
- Subjects :
- a priori estimates
fast diffusion equation
Type (model theory)
01 natural sciences
nonlocal operators
Mathematics - Analysis of PDEs
porous medium equation
Convergence (routing)
FOS: Mathematics
Uniqueness
Mathematics - Numerical Analysis
0101 mathematics
Mathematics
nonlinear degenerate diffusion
Numerical Analysis
convergence
Applied Mathematics
Numerical analysis
010102 general mathematics
Mathematical analysis
Stefan problem
existence
uniqueness
Numerical Analysis (math.NA)
010101 applied mathematics
Computational Mathematics
Nonlinear system
Fully discrete numerical schemes
fractional Laplacian
Laplacian
Porous medium
Laplace operator
distributional solutions
Analysis of PDEs (math.AP)
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- BIRD: BCAM's Institutional Repository Data, instname, SIAM Journal on Numerical Analysis
- Accession number :
- edsair.doi.dedup.....56165efbd314db6161c1162b1243c874