Abstract
In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.
Highlights
Received: 26 October 2021Accepted: 13 November 2021Published: 18 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil-In this paper, we are interested in developing efficient numerical solutions of strongly nonlinear space-fractional diffusion equations (NSFDEs) of the following form: ∂u( x,t) ∂α u( x,t)( x, t) ∈ (0, L x ) × (0, T ],∂t = a(u) ∂| x|α + f ( x, t), u( x, 0) = φ( x ), u(0, t) = φ(t), u( L, t) = ψ(t), x x ∈ [0, L x ], (1)t ∈ [0, T ], iations
We are interested in developing efficient numerical solutions of strongly nonlinear space-fractional diffusion equations (NSFDEs) of the following form:
We study the efficient numerical solution of strongly NSFDEs because they are more suitable than linear models to describe some difficult physical processes, such as fractional distillation in nonlinear space, and when the diffusion coefficient is related to the concentration of a particle plume [31]
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil-. In contrast with the classical diffusion operator ∆, Riesz fractional derivative [3,5] is a special linear combination of left- and right-sided Riemann–Liouville fractional derivatives By exploiting this property, Ding and Li [4] established a novel class of high-order numerical algorithms for Riesz derivatives through constructing new generating functions. We study the efficient numerical solution of strongly NSFDEs because they are more suitable than linear models to describe some difficult physical processes, such as fractional distillation in nonlinear space, and when the diffusion coefficient is related to the concentration of a particle plume [31]. Our main contribution to the subject can be summarized as follows: first, for this class of strongly NSFDEs, we propose a semi-implicit difference scheme that can avoid solving the discretized nonlinear systems [35], and analyze its stability and convergence properties. The proposed method uses a fractional central difference formula to discretize the Riesz fractional derivative; it is stable and achieves first-order convergence in time and second-order convergence in space, respectively
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