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A novel fourth-order scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations and its optimal preconditioned solver

Wei Qu, Yuan-Yuan Huang, Sean Hon, Siu-Long Lei

TL;DR

The paper tackles 2D Riesz space fractional nonlinear reaction-diffusion equations with orders $1<\alpha,\beta<2$ and Lipschitz nonlinearity, introducing a novel fourth-order finite difference scheme coupled with a Crank-Nicolson explicit linearization. It proves unconditional stability and convergence in the discrete $L_2$-norm, and develops a $\tau$-matrix based preconditioner that diagonalizes via DST to deliver mesh-independent PCG convergence for the resulting symmetric Toeplitz systems. The preconditioned system has eigenvalues strictly bounded in $(3/8,2)$, ensuring linear PCG convergence independent of mesh size, and the approach outperforms circulant preconditioners in numerical tests. Numerical experiments confirm high accuracy and substantial efficiency gains, highlighting the method’s practical impact for multi-dimensional fractional PDEs and scalable solvers.

Abstract

A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high-order scheme is proved to be unconditionally stable and convergent in the discrete $L_2$-norm. Moreover, a $τ$-matrix based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(3/8,2)$. To the best of our knowledge, this is the first attempt to develop a preconditioned iterative solver with a mesh-independent convergence rate for the linearized high-order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.

A novel fourth-order scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations and its optimal preconditioned solver

TL;DR

The paper tackles 2D Riesz space fractional nonlinear reaction-diffusion equations with orders and Lipschitz nonlinearity, introducing a novel fourth-order finite difference scheme coupled with a Crank-Nicolson explicit linearization. It proves unconditional stability and convergence in the discrete -norm, and develops a -matrix based preconditioner that diagonalizes via DST to deliver mesh-independent PCG convergence for the resulting symmetric Toeplitz systems. The preconditioned system has eigenvalues strictly bounded in , ensuring linear PCG convergence independent of mesh size, and the approach outperforms circulant preconditioners in numerical tests. Numerical experiments confirm high accuracy and substantial efficiency gains, highlighting the method’s practical impact for multi-dimensional fractional PDEs and scalable solvers.

Abstract

A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high-order scheme is proved to be unconditionally stable and convergent in the discrete -norm. Moreover, a -matrix based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval . To the best of our knowledge, this is the first attempt to develop a preconditioned iterative solver with a mesh-independent convergence rate for the linearized high-order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.
Paper Structure (6 sections, 10 theorems, 96 equations, 3 tables)

This paper contains 6 sections, 10 theorems, 96 equations, 3 tables.

Table of Contents

  1. Introduction
  2. A fourth-order approximation to the Riesz fractional derivative
  3. Fully discrete numerical scheme and its theoretical analysis
  4. $\tau$-matrix based preconditioner and its spectral analysis
  5. Numerical experiments
  6. Concluding remarks

Key Result

Lemma 2.1

(ding2023high) Let be the fractional Sobolev space. Suppose $u(x)\in\mathcal{L}^{4+\mu}(\mathbb{R})$, then we have the following fourth-order approximation: where the coefficients $s_{k}^{(\alpha)}$$(k=0,\pm 1,\pm 2,\ldots)$ are determined by the Fourier expansion of the generating function Setting $\omega=e^{{\bf i}\theta}$ with ${\bf i}=\sqrt{-1}$ in Eq. (generatingS) and using the inverse Fo

Theorems & Definitions (10)

  • Lemma 2.1
  • Proposition 1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Theorem 4.1