Finite volume schemes for multilayer diffusion

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:345

Article

March, Nathan G.¹  Carr, Elliot J.¹ 

[1] Queensland Univ Technol, Sch Math Sci, Brisbane, Qld, Australia

关键词: Multilayer diffusion;    Finite volume scheme;    Stability and convergence;    Gershgorin circle theorem;    Interface conditions;   

DOI  :  10.1016/j.cam.2018.06.041

来源: Elsevier

【 摘 要 】

This paper focusses on finite volume schemes for solving multilayer diffusion problems. We develop a finite volume method that addresses a deficiency of recently proposed finite volume/difference methods, which consider only a limited number of interface conditions and do not carry out stability or convergence analysis. Our method also retains secondorder accuracy in space while preserving the tridiagonal matrix structure of the classical single-layer discretisation. Stability and convergence analysis of the new finite volume method is presented for each of the three classical time discretisation methods: forward Euler, backward Euler and Crank-Nicolson. We prove that both the backward Euler and Crank-Nicolson schemes are always unconditionally stable. The key contribution of the work is the presentation of a set of sufficient stability conditions for the forward Euler scheme. Here, we find that to ensure stability of the forward Euler scheme it is not sufficient that the time step r satisfies the classical constraint of tau <= h(i)(2)/(2D(i)) in each layer (where D i is the diffusivity and hi is the grid spacing in the ith layer) as more restrictive conditions can arise due to the interface conditions. The paper concludes with some numerical examples that demonstrate application of the new finite volume method, with the results presented in excellent agreement with the theoretical analysis. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free