Preconditioned quasi-compact boundary value methods for space-fractional diffusion equations

Abstract

This paper focuses on highly-efficient numerical methods for solving spacefractional diffusion equations. By combining the fourth-order quasi-compact difference scheme and boundary value methods, a class of quasi-compact boundary value methods are constructed. In order to accelerate the convergence rate of this class of methods, the Kronecker product splitting (KPS) iteration method and the preconditioned method with KPS preconditioner are proposed. A convergence criterion for the KPS iteration method is derived. A numerical experiment further illustrates the computational efficiency and accuracy of the proposed methods. Moreover, a numerical comparison with the preconditioned method with Strang-type preconditioner is given, which shows that the preconditioned method with KPS preconditioner is comparable in computational efficiency.

Compact block boundary value methods for semi‐linear delay‐reaction–diffusion equations with algebraic constraints

“…For the spatial discretization, compact difference methods have been used successfully to solve some PDEs without algebraic constraint (cf. [6, 11–13, 16–20]). For the temporal discretization, a good candidate is block boundary value methods (BBVMs) due to their excellent stability and high accuracy.…”

“…We note that, among the efficient numerical methods applied to some partial differential equations (PDEs), there are a class of so‐called linear approximation methods (cf. [8–11, 13, 14, 16]), where the spatial and temporal variables are discretized, respectively. Comparing with the time–space fully discretization methods, linear approximation methods can flexibly combine some spatial and temporal methods such that the whole methods arrive at the advantages of high‐efficiency and high‐accuracy.…”

“…For the temporal discretization, a good candidate is block boundary value methods (BBVMs) due to their excellent stability and high accuracy. For example, in References [14–16, 21–28], BBVMs were applied to ordinary differential equations (ODEs), differential‐algebraic equations, delay/functional differential equations and the temporal discretization of semi‐linear parabolic equations, space‐fractional diffusion equations and delay reaction–diffusion equations.…”

An implicit difference scheme with the KPS preconditioner for two-dimensional time–space fractional convection–diffusion equations

“…In addition, some other accelerating methods have been also presented, such as the multigrid method, the fast ADI method, the preconditioned conjugate gradient method, the multilevel circulant preconditioned method and the Kronecker product splitting (KPS) method (cf. [22][23][24][25][26][27][28]).…”

Fast IMEX Time Integration of Nonlinear Stiff Fractional Differential Equations

“…More recently, Garrappa [16] developed trapezoidal methods for fractional multi-step approaches and Zeng [58] developed a second-order scheme for time-fractional diffusion equations. Spectral methods were also developed in the context of FDEs/FPDEs [21,23,37,39,40,45,[54][55][56]67], and distributed-order differential equations [20,22,38]. In particular, Zayernouri and Karniadakis [55] developed an exponentially-accurate spectral element method for FDEs and Lischke et al [25] developed a fast, tunably-accurate spectral method.…”