Robust Numerical Methods for Nonlocal (and Local) Equations of Porous Medium Type. Part II: Schemes and Experiments

Abstract

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − L[ϕ(u)] = f (x, t) in R N × (0, T), where L is a general symmetric Lévy type diffusion operator. Included are both local and nonlocal problems with e.g. L = ∆ or L = −(−∆) α 2 , α ∈ (0, 2), and porous medium, fast diffusion, and Stefan type nonlinearities ϕ. 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 ∈ [1, ∞], compact, and convergent in C([0, T ]; L p loc (R N)) for p ∈ [1, ∞). The first part of this project is given in [36] 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 [36] 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.

A Finite Difference Method for the Variational p-Laplacian

“…The drawback of this type of regularization is that the condition (CFL) becomes more and more restrictive as δ → 0. This regularization is typically used when dealing with explicit schemes for fast diffusion equations (see for example [8,9])…”

On an Anisotropic Fractional Stefan-Type Problem with Dirichlet Boundary Conditions

“…Although in general η, ϑ may be nonzero outside Ω, except for the bilinear form ´Rd ×[0,T ] AD s ϑ • D s ξ, the other integral terms in the variational formulation (1.14) are only integrated over Ω in space, since the test function ξ is 0 in Ω c ×]0, T [. Different non-local versions of Stefan-type problems have previously been considered, including in [10] and [16] for nonsingular integral kernels, in [53], [8], [44] and [42] for the fractional Caputo derivatives, and in [22], [23], [24], [25] and [30] for the fractional Laplacian and its nonlocal integral generalization in [2]. Stefan-type problems that are fractional in the time derivative have also been considered (see, for instance, [43], [34] and [15].)…”

“…Stefan-type problems that are fractional in the time derivative have also been considered (see, for instance, [43], [34] and [15].) Indeed, when the matrix A is a multiple of the identity matrix, the fractional Stefan-type problem (1.2) reduces to that with the fractional Laplacian as considered in [22]- [25]. Furthermore, in instances as described in Section 2.3 of [36] when the fractional operator L s A is replaced with a nonlocal operator Ls a , corresponding to a Dirichlet form with the kernel a which satisfies some compatibility conditions, (1.2) may also be considered a nonlocal Stefan problem, as considered in [2].…”

Fast implicit difference schemes for time‐space fractional diffusion equations with the integral fractional Laplacian

“…[3][4][5][6] A physical-mathematical model to anomalous diffusion may be based on FPDEs containing derivatives of fractional order in both space and time, where the subdiffusion appears in time and the superdiffusion occurs in space simultaneously. 7,8 On the other hand, although most of time-space fractional diffusion models are initially defined with the spatially integral fractional Laplacian (IFL), [9][10][11][12] many previous studies (cf., e.g., previous works 4,5,[13][14][15] ) always substitute the space Riesz fractional derivative 1 for the IFL. In fact, such two kinds of definitions are not equivalent in high-dimensional cases.…”

“…3,[19][20][21][22][23] Due to the nonlocality, the analytical (or closed-form) solutions of TSFDEs (1.1) on a finite domain are rarely available. Therefore, we have to rely on numerical treatments that produce approximations to the desired solutions; refer, for example, to previous studies 1,12,[24][25][26] and references therein for a description of such approaches. In fact, utilizing the suitable temporal discretization, most of the early established numerical methods including the finite difference (FD) method, 8,[27][28][29] finite element (FE) method, 30,31 and matrix (named it as all-at-once) method 13,32 for the TSFDE (1.1) were developed via the fact that the IFL is equivalent to the Riesz fractional derivative in one space dimension.…”