Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation

Journal of Computational Physics

et al. 2019

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In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen-Cahn equations with smooth and non-smooth solutions. The implicit second-order θ scheme containing both implicit Crank-Nicolson scheme and second-order backward difference method is applied to time direction, a fast TT-M method is used to increase the speed of calculation, and the FE method is developed to approximate the spacial direction. The TT-M FE algorithm includes the following main computing steps: firstly, a nonlinear implicit second-order θ FE scheme on the time coarse mesh τ c is solved by a nonlinear iterative method; secondly, based on the chosen initial iterative value, a linearized FE system on time fine mesh τ < τ c is solved, where some useful coarse numerical solutions are found by the Lagrange's interpolation formula. The analysis for both stability and a priori error estimates are made in detail. Finally, three numerical examples with smooth and non-smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations, from which it is easy to find that the computing time can be saved. RL D α c,y u = 1 Fast TT-M FE algorithm combined with second-order θ scheme is used to solve the nonlinear space 1 2 , (2.5) and norm u J β S (Ω) = ( u 2 + |u| 2 J β S (Ω) ) 1 2 , (2.6)and denote by J β S (Ω)(or J β S,0 (Ω)) the closure of C ∞ (Ω)(or C ∞ 0 (Ω)) with respect to · J β S (Ω) .

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast algorithm based on TT-M FE system for space fractional Allen–Cahn equations with smooth and non-smooth solutions

Yin

Journal of Computational Physics

et al. 2019

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“…To formulate the time discrete scheme, referring to Ref. [23], we have the following lemmas at time t = t n−θ . Lemma 2.1 For sufficiently smooth function φ(t), at time t n−θ , the following approximation for first-order derivative with second-order convergence rate for any θ ∈ [0, 1 2…”

Section: Numerical Schemementioning

confidence: 99%

TGMFE algorithm combined with some time second-order schemes for nonlinear fourth-order reaction diffusion system

Yin

Results in Applied Mathematics

et al. 2019

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In this article, a two-grid mixed finite element (TGMFE) method with some second-order time discrete schemes is developed for numerically solving nonlinear fourth-order reaction diffusion equation. The two-grid MFE method is used to approximate spatial direction, and some second-order θ schemes formulated at time t k−θ are considered to discretize the time direction. TGMFE method covers two main steps: a nonlinear MFE system based on the space coarse grid is solved by the iterative algorithm and a coarse solution is arrived at, then a linearized MFE system with fine grid is considered and a TGMFE solution is obtained. Here, the stability and a priori error estimates in L 2 -norm for both nonlinear Galerkin MFE system and TGMFE scheme are derived. Finally, some convergence results are computed for both nonlinear Galerkin MFE system and TGMFE scheme to verify our theoretical analysis, which show that the convergence rate of the time second-order θ scheme including Crank-Nicolson scheme and second-order backward difference scheme is close to 2, and that with the comparison to the computing time of nonlinear Galerkin MFE method, the CPU-time by using TGMFE method can be saved.

“…In fact, many mathematical models and problems from science and engineering must be computed on irregular domains and therefore seeking effective numerical methods to solve these problems on such domains is important. Although existing numerical methods for fractional diffusion equations are numerous [23,24,25,26,27,28,29,30,31,32,33,34], most of them are limited to regular domains and uniform meshes. Research involving unstructured meshes and irregular domains is sparse.…”

Section: Introductionmentioning

confidence: 99%

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

Feng

Journal of Computational and Applied Mathematics

Turner

2020

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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 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.