Numerical Method for the Time-Fractional Porous Medium Equation

Abstract

This papers deals with a construction and convergence analysis of a finite difference scheme for solving time-fractional porous medium equation. The governing equation exhibits both nonlocal and nonlinear behaviour making the numerical computations challenging. Our strategy is to reduce the problem into a single one-dimensional Volterra integral equation for the selfsimilar solution and then to apply the discretization. The main difficulty arises due to the non-Lipschitzian behaviour of the equation's nonlinearity. By the analysis of the recurrence relation for the error we are able to prove that there exists a family of finite difference methods that is convergent for a large subset of the parameter space. We illustrate our results with a concrete example of a method based on the midpoint quadrature.for the main operator in the spatial diffusion, namely the fractional Laplacian, can be found in [20,11]. As for the nonlinear version of the anomalous diffusion, an interesting paper recently appeared in which the authors constructed a multi-grid waveform method of fast convergence [18]. Moreover, a similar equation to ours has been solved in [3] in the context of petroleum industry. Some other numerical approaches concern space-fractional nonlinear diffusion [17,14], nonlinear source terms [40,38,5] and variable order diffusion [26].Our approach is based on a transformation of the governing PDE to the equivalent nonlinear Volterra equation (for a recent survey of theory and numerical methods see [6]). In that case the classical convergence theory cannot be applied since the nonlinearity of the equation is non-Lipschitzian. According to our best knowledge, there is a scarce literature concerning similar problems. A very interesting paper is [7] where an iterative technique has been applied and convergence proofs given. Moreover, in [8] a short summary of the theoretical and numerical character has been published. Lastly, we mention our own work [31] from which the present considerations stem where we have given the convergence proof assuming the kernel's separation from zero. In the present discussion we relax this assumption.The paper is structured as follows. In the second section we formulate the problem in terms of the Volterra setting starting from the self-similar form of the time-fractional porous medium equation. We proceed to the main results in the third section where a construction of a convergent finite difference method is given. We end the paper with some numerical simulations illustrating our results.

How to Define Dissipation-Preserving Energy for Time-Fractional Phase-Field Equations

“…It is known that the historic memory of time-fraction plays a significant role as demonstrated in many numerical simulations, see, e.g., [16][17][18]. Although the whole evolution process may be slower due to the memory effect, it is still expected that main regularity properties, nonlinear stability and other main features of the relevant phase-field equations will be preserved.…”

Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line

“…This method could be proved to be convergent (see [55]), however, it does not solve the constant case exactly, that is to say when K(z, s) = K + (z − s) γ the numerical solution v n is not equal to (45).…”

“…However, we have some useful asymptotics. First, let us consider the Dirichlet boundary condition for which we know that (see [55])…”

“…In previous works we have proved existence and uniqueness of a self-similar solution to the Dirichlet problem [57], provided some approximations [51], solved inverse problem of diffusivity identification [53], and constructed a numerical scheme [55]. This numerical method, although only first order accurate, provided a superior performance over the finite difference method applied to the governing PDE.…”

Alternative probabilistic representations of Barenblatt-type solutions

“…On the other hand, a probabilistic approach to time-fractional nonlinear diffusivetype equations is still completly missing. Recently, the existence and uniqueness of compactly supported solutions for time-fractional porous medium equations has been investigated (see, e.g., [30]). However, up to our knowledge, it is not possible to find an explicit form of the Barenblatt-type solution.…”