High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fract
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High‑Order Local Discontinuous Galerkin Algorithm with Time Second‑Order Schemes for the Two‑Dimensional Nonlinear Fractional Diffusion Equation Min Zhang1 · Yang Liu2 · Hong Li2 Received: 21 October 2019 / Revised: 9 December 2019 / Accepted: 13 December 2019 © Shanghai University 2020
Abstract In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order 𝜃 approximation formulas in time are presented to solve a two-dimensional nonlinear fractional diffusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with O(hk+1 + 𝛥t2 ) is derived, where k ⩾ 0 denotes the index of the basis function. Extensive numerical results with Qk (k = 0, 1, 2, 3) elements are provided to confirm our theoretical results, which also show that the secondorder convergence rate in time is not impacted by the changed parameter 𝜃. Keywords Two-dimensional nonlinear fractional diffusion equation · High-order LDG method · Second-order 𝜃 scheme · Stability and error estimate Mathematics Subject Classification 65M60 · 65N30
1 Introduction In this article, we consider the two-dimensional nonlinear fractional diffusion model
𝜕u(x, y, t) R 𝛼 +0 Dt u(x, y, t) − Δu(x, y, t) + f (u) = g(x, y, t), (x, y, t) ∈ 𝛺 × (0, T] 𝜕t
(1)
with the periodic boundary condition and the initial condition
u(x, y, 0) = u0 (x, y),
(2)
̄ (x, y) ∈ 𝛺,
where 𝛺 ⊂ ℝ2 and (0, T] are the space domain and the time interval with 0 < T < ∞ , respectively. The source term g(x, y, t) and the initial value u0 (x, y) are given functions. The nonlinear function f(u) satisfies |f (u)| ⩽ C|u| with |f � (u)| ⩽ C , where C is a positive
* Yang Liu [email protected] 1
School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian Province, China
2
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
13
Vol.:(0123456789)
Communications on Applied Mathematics and Computation
constant. The R0 D𝛼t z(t) is the Riemann–Liouville time fractional derivative with 𝛼 ∈ (0, 1) , which is given by t
R 𝛼 D z(t) 0 t
=
z(s) 𝜕 1 ds. Γ(1 − 𝛼) 𝜕t ∫0 (t − s)𝛼
(3)
Fractional partial differential equations (FPDEs) are very important mathematical models, which have played a very important role in science and engineering fields [1, 12, 14, 17, 26, 29, 31]. It is more suitable to describe some systems with memory and hereditary than integer-order ones due to its remarkable performance and hereditary properties. So far, fractional diffusion models have been solved numerically and analytically in increasing papers [4, 10, 11, 15, 18–20, 22–24, 30, 32, 39–47]. Tian et al. [34] proposed secondorder WSGD approximations for space fractional diffusion equations. Based on the idea in [34], Wang and Vong [35] developed the WSGD formulas for time fractional derivatives and gave some compact difference schemes for the fractional subdiffusion problem and diffusion-wave equation with the time fractional derivative term. In [21], Liu et al. considered the fast two-
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