NUMERICAL METHOD FOR THE TIME-FRACTIONAL POROUS MEDIUM EQUATION

摘要

This paper deals with a construction and convergence analysis of a numerical scheme devised for solving the time-fractional porous medium equation with Dirichlet boundary conditions on the half line. The governing equation exhibits both nonlocal and nonlinear behavior making the numerical computations challenging. Our strategy is to reduce the problem into a single one-dimensional Volterra integral equation for the self-similar solution and then to apply a suitable discretization. The main difficulty arises due to the non-Lipschitzian behavior of the nonlinearity of the corresponding integral equation. By the analysis of the recurrence relation for the error, we are able to prove that there exists a family of schemes that is convergent for a large subset of the parameter space. More specifically, in the very slow regime of the subdiffusion, we require that the diffusivity parameter has to be sufficiently large to provide the convergence of the method. We illustrate our results with a concrete example of a method based on the midpoint quadrature.