An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:364

Article

Feng, Libo¹  Liu, Fawang^1,2  Turner, Ian^1,3 

[1] Queensland Univ Technol, Sch Math Sci, GPO Box 2434, Brisbane, Qld 4001, Australia

[2] Fuzhou Univ, Coll Math & Comp Sci, Fuzhou 350116, Fujian, Peoples R China

[3] QUT, Australian Res Council Ctr Excellence Math & Stat, Brisbane, Qld, Australia

关键词: Control volume method;    Unstructured mesh;    Fast iterative solver;    Space fractional derivative;    Irregular convex domains;    Two-dimensional;   

DOI  :  10.1016/j.cam.2019.06.035

来源: Elsevier

【 摘 要 】

In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature. Firstly, we present the finite volume scheme for the two-dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements. Secondly, we explore the property of the stiffness matrix generated by the integral of the space fractional derivative. We find that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system, which is more efficient than using the Gaussian elimination method. Finally, we present several examples to verify our method, in which we make a comparison of our method with the finite element method for solving a Riesz space fractional diffusion equation on a circular domain. The numerical results demonstrate that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. The numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

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