Sparse Monte Carlo method for nonlocal diffusion problems. (English) Zbl 1502.65125
A numerical method for solving nonlocal, nonlinear diffusion equations like
\[
u_t(t,x)=\int W(x,y)D(u(t,y)-u(t,x))dy+f(u,x,t)
\]
is proposed. The motivation for study such equations stems from modeling propagation phenomena in continuous media with nonlocal interactions. This numerical method combines sparse Monte Carlo and discontinuous Galerkin methods. Under suitable assumptions on the kernel \(W\), Lipschitz continuity of the function \(D\) and “nonhomogeneous” term \(f\), convergence of the proposed numerical method is shown together with estimations of errors. Numerical examples are provided.
Reviewer: Piotr Biler (Wrocław)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 65C05 | Monte Carlo methods |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
Keywords:
nonlinear diffusion; nonlocal diffusion; initial value problem; sparse Monte Carlo method; error estimateReferences:
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