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Finite difference method for a fractional porous medium equation. (English) Zbl 1310.76115

Summary: We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[ \frac{\partial u}{\partial t}+(-\Delta)^{1/2} (u^m)=0. \] The problem is posed in \(x\in \mathbb R^N\), \(m\geq 1\) and with nonnegative initial data. The fractional Laplacian is implemented via the so-called Caffarelli-Silvestre extension. We prove existence and uniqueness of the solution of this method and also the convergence to the theoretical solution of the equation. We run numerical experiments on typical initial data as well as a section that summarizes and concludes the proposed method.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K59 Quasilinear parabolic equations
35R11 Fractional partial differential equations
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
76S05 Flows in porous media; filtration; seepage
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References:

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