Second-Order Finite Difference/Spectral Element Formulation for Solving the Fractional Advection-Diffusion Equation
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Second‑Order Finite Difference/Spectral Element Formulation for Solving the Fractional Advection‑Diffusion Equation Mostafa Abbaszadeh1 · Hanieh Amjadian1 Received: 14 August 2019 / Revised: 12 January 2020 / Accepted: 20 January 2020 © Shanghai University 2020
Abstract The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equation. The time variable has been discretized by a second-order finite difference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confirm the theoretical results. Keywords Spectral method · Finite difference method · Fractional advection-diffusion equation · Galerkin weak form · Unconditional stability Mathematics Subject Classification 65L60 · 65L20 · 65M70
1 Introduction The fractional advection-diffusion is one of the important models in the fractional PDEs [28, 44]. The fractional advection-diffusion has been solved by several numerical methods such as the operational matrix approach [7, 47], the finite difference method [11], the finite element method [9], the spectral collocation techniques [4, 46], some high-order numerical approximations [30], the ADI meshless method [2], etc. In addition, the interested readers can refer to [1, 6, 16, 17, 20, 22, 48] to find more information for the numerical solutions of fractional PDEs. The main purpose of the current paper is to study the fractional Galilei invariant advection-diffusion equation
* Mostafa Abbaszadeh [email protected] Hanieh Amjadian [email protected] 1
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran 15914, Iran
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Vol.:(0123456789)
Communications on Applied Mathematics and Computation
⎧ 𝜕u + ∇u − Δu = 0 D1−𝛾 t (Δu) + f (𝐱, t), ⎪ 𝜕t ⎪ ⎨ u(𝐱, t) = 0, 𝐱 = (x, y) ∈ 𝜕Ω, ⎪ ⎪ ⎩ u(𝐱, 0) = u0 (𝐱), 𝐱 = (x, y) ∈ Ω,
𝐱 = (x, y) ∈ Ω,
0 < 𝛾 < 1, (1.1)
in which Ω is a bounded convex domain in the plane, 𝜕Ω is the Lipschitz continuous is the boundary of Ω , and 𝐱 is a vector of the independent variables. The symbol 0 D1−𝛾 t Riemann–Liouville fractional derivative operator and is defined as t
1−𝛾 0 Dt u(x, t)
=
u(x, 𝜂) 1 𝜕 d𝜂, Γ(𝛾) 𝜕t ∫0 (t − 𝜂)1−𝛾
where Γ(.) is the gamma function. The high-order numerical procedures based upon the spectral idea have been investigated for solving the factional PDEs, for example the cable equation [34], the Fokker–Planck model [52], the advection-diffusion models [5, 49], the Schrödinger model [50, 51], the coupled reaction-advection-diffusion equations [38], the FitzHugh–Nagumo monodomain model [8], the fractional advection-dispersion equation [26], the vari
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