A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations
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A Local Discontinuous Galerkin Method for Two‑Dimensional Time Fractional Diffusion Equations Somayeh Yeganeh1 · Reza Mokhtari1 · Jan S. Hesthaven2 Received: 22 July 2019 / Revised: 10 March 2020 / Accepted: 14 March 2020 © Shanghai University 2020
Abstract For two-dimensional (2D) time fractional diffusion equations, we construct a numerical method based on a local discontinuous Galerkin (LDG) method in space and a finite difference scheme in time. We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable. Numerical results indicate the effectiveness and accuracy of the method and confirm the analysis. Keywords Two-dimensional (2D) time fractional diffusion equation · Local discontinuous Galerkin method (LDG) · Numerical stability · Convergence analysis Mathematics Subject Classification 65M60 · 65M12
1 Introduction Time fractional diffusion equations are obtained from the standard diffusion equations by replacing the first-order time derivative with a fractional derivative of order 𝛼 ∈ (0, 2) . In this paper, we consider the following time-fractional diffusion equation:
⎧ D𝛼 u(x, t) − Δu(x, t) = f (x, t), x = (x, y) ∈ Ω, t ∈ (0, T], ⎪ t (x, y) ∈ Ω, ⎨ u�t=0 = Φ(x, y), ⎪ u(x, y, t) = Ψ(x, y, t), (x, y) ∈ 𝜕Ω, t ∈ (0, T], ⎩
(1)
where Φ , Ψ , and f are given functions, Ω ⊂ ℝ2 is a bounded rectangular domain with boundary 𝜕Ω , and D𝛼t is the Caputo fractional derivative defined as
* Reza Mokhtari [email protected] 1
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156‑83111, Iran
2
EPFL-SB-MATHICES-MCSS, École Polytechnique Fédéral de Lausanne, 1015 Lausanne, Switzerland
13
Vol.:(0123456789)
Communications on Applied Mathematics and Computation
D𝛼t u(x, t) =
t 𝜕u(x, s) ds 1 , ∫ 𝚪(1 − 𝛼) 0 𝜕s (t − s)𝛼
(2)
0 < 𝛼 < 1,
in which 𝚪 is the Gamma function. For 𝛼 = 1 , we have D𝛼t u = ut. Diffusion equations of fractional order are used to describe anomalous diffusion in transport processes, see, e.g., [30], and monographs [31, 33]. Recently, some handbooks related to fractional calculus and its applications in physics, engineering, and life science as well as social science have been published. Interested readers find some basic theories in volume 1 [22] of the handbooks and some other materials can be found in other volumes. Analytic solutions for most fractional partial differential equations (FPDEs) are complicated to obtain or cannot be expressed explicitly. Hence, methods for finding numerical solutions for such problems are required. Numerical methods proposed for solving the FPDEs include finite difference methods [1, 6, 9, 11, 15, 25, 28, 29, 45], finite element methods [12, 14, 19, 20, 36, 44], the finite volume method [21], spectral methods [2, 24, 26], discontinuous Galerkin methods [7, 8, 13, 39, 41, 42], and homotopy and variational methods [17, 34, 40, 43]. The first local discontinuous Galerkin (LDG) method was introduced by Cockburn and Shu in [5] f
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