Efficient Numerical Schemes and Fast Preconditioners for Riesz Fractional Diffusion Equations with Variable Coefficients

Core Concepts

This study proposes an efficient numerical scheme and a fast sine transform-based preconditioner to solve Riesz fractional diffusion equations with variable coefficients.

Abstract

The key highlights and insights of this content are: The authors consider the numerical solution of Riesz space fractional diffusion equations (RSFDEs) with variable coefficients in multiple dimensions. They develop a fourth-order quasi-compact finite difference scheme to discretize the temporal and spatial fractional derivatives. This results in a linear system with a coefficient matrix that is the sum of a product of a (block) tridiagonal matrix and a diagonal matrix, and a d-level Toeplitz matrix. To accelerate the convergence of the GMRES method for solving the resulting linear systems, the authors propose a sine transform-based preconditioner. This preconditioner is shown to be symmetric positive definite. Theoretical analysis demonstrates that the preconditioned GMRES method has a convergence rate that is independent of the grid size, fractional order, and spatial dimension. Numerical experiments confirm the theoretical results and illustrate the efficiency of the proposed preconditioner compared to the one-sided preconditioned GMRES method.

Stats

The content does not provide specific numerical data to extract. The focus is on the theoretical analysis and development of the numerical scheme and preconditioner.

Quotes

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Key Insights Distilled From

On $τ$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients

by Zi-Hang She,... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10221.pdf

Deeper Inquiries

How can the proposed preconditioner be extended to handle more general types of coefficient matrices beyond the Toeplitz-like structure considered in this work

The proposed preconditioner based on the fast discrete sine transform can be extended to handle more general types of coefficient matrices beyond the Toeplitz-like structure considered in this work by incorporating additional matrix algebra techniques. One approach could be to explore the use of block matrix preconditioning methods, such as block diagonal or block triangular preconditioners, to address more complex coefficient structures. By decomposing the coefficient matrix into blocks and applying appropriate preconditioning strategies to each block, the preconditioner can be tailored to handle a wider range of coefficient matrix configurations. Additionally, exploring the use of domain decomposition methods in conjunction with the sine transform-based preconditioner can also enhance its applicability to more general coefficient matrices.

What are the potential limitations or challenges in applying the sine transform-based preconditioner to high-dimensional fractional diffusion problems with non-constant coefficients

One potential limitation in applying the sine transform-based preconditioner to high-dimensional fractional diffusion problems with non-constant coefficients is the computational complexity associated with the discretization and preconditioning of dense matrices. As the dimensionality of the problem increases, the size of the coefficient matrices grows significantly, leading to higher computational costs for matrix operations and preconditioning. Moreover, handling non-constant coefficients introduces additional challenges in accurately approximating the preconditioner and ensuring its effectiveness in accelerating the convergence of iterative solvers. Balancing the trade-off between accuracy and efficiency in preconditioning high-dimensional problems with varying coefficients remains a key challenge in applying the sine transform-based approach.

Can the ideas behind the proposed preconditioner be adapted to develop efficient solvers for other types of fractional partial differential equations beyond the Riesz fractional diffusion equation

The ideas behind the proposed preconditioner can be adapted to develop efficient solvers for other types of fractional partial differential equations beyond the Riesz fractional diffusion equation by customizing the preconditioning strategies to suit the specific characteristics of the equations. For instance, for fractional advection-diffusion equations or fractional wave equations, the preconditioner can be tailored to incorporate the specific fractional derivative operators and spatial discretization schemes used in those equations. Additionally, exploring the use of hybrid preconditioning techniques, such as combining the sine transform-based preconditioner with domain decomposition methods or multigrid solvers, can further enhance the efficiency of the solvers for a broader class of fractional PDEs. By adapting the underlying principles of the proposed preconditioner to different types of fractional PDEs, more effective and versatile numerical solvers can be developed.