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

Yin, Baoli; Liu, Yang; Li, Hong; He, Shuping

doi:10.1016/j.jcp.2018.12.004

Cited by 63 publications

(12 citation statements)

References 49 publications

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“…For fractional calculus equations, one generally can not get the solution in closed form, thus different numerical methods have been proposed to efficiently obtain the approximate solution, see [2,5,7,8,10,20,21,24,25,31,38,41,43,49]. As is well known that the solution of fractional differential equations shows some singularity at the initial node, different techniques were employed to restore the optimal convergence rate, see [15,23,26,27,40,42,44,45,47].…”

Section: Baoli Yin Yang Liu Hong LI and Zhimin Zhangmentioning

confidence: 99%

Approximation methods for the distributed order calculus using the convolution quadrature

Yin¹,

Liu²,

Li³

et al. 2021

Discrete &Amp; Continuous Dynamical Systems - B

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In this article we generalize the convolution quadrature (CQ) method, which aims at approximating the fractional calculus, to the case for the distributed order calculus. Our method is a natural expansion that the approximation formulas, convergence results and correction technique reduce to the cases for the CQ method if the weight function µ(α) is defined by δ(α − α 0 ). Further, we explore a new structure of the solution of an ODE with the distributed order fractional derivative, which differs from those of the solutions of traditional fractional ODEs, and propose a new correction technique for this new structure to restore the optimal convergence rate. Numerical tests with smooth and nonsmooth solutions confirm our theoretical results and the efficiency of our correction technique.

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Section: Baoli Yin Yang Liu Hong LI and Zhimin Zhangmentioning

confidence: 99%

Approximation methods for the distributed order calculus using the convolution quadrature

Yin¹,

Liu²,

Li³

et al. 2021

Discrete &Amp; Continuous Dynamical Systems - B

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“…To give the error analysis, we need to introduce a projection operator

P_{normalℏ} : H_{0}^{η} false(normalΩ false) \to V_{normalℏ}, u \in H_{0}^{η} false(normalΩ false)

satisfying

B false(P_{normalℏ} u, v false) = B false(u, v false), \forall v \in V_{normalℏ},

where ℏ is taken as h or H . Furthermore, we have the following lemma.Lemma (See [39, 40]) If

u \in H^{μ} false(normalΩ false) \cap H_{0}^{η} false(normalΩ false), η < μ \leq r + 1

, we have

‖ P_{h} u - u ‖_{η} \leq C {normalℏ}^{μ - η} ‖ u ‖_{μ} .

Based on the relation (3.14) and Lemma 3.3, we will derive the convergence.Theorem With

u false(t_{n} false) \in H^{μ} false(normalΩ false) \cap H_{0}^{η} false(normalΩ false)

and μ = r + 1, we obtain the following a priori error result

‖ u false(t_{n} false) - u_{h}^{n} ‖ + {((), normalΔ t \sum_{k = 1}^{n} ‖ u (t_{k}) - u_{h}^{k} ‖_{η}^{2})}^{\frac{1}{2}} \leq C false(Δ t^{2} + h^{r + 1 - η}

…”

Section: Stability and Error Estimatesmentioning

confidence: 99%

Fast calculation based on a spatial two‐grid finite element algorithm for a nonlinear space–time fractional diffusion model

Yang

Liu

et al. 2020

Numerical Methods Partial

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In this article, a spatial two-grid finite element (TGFE) algorithm is used to solve a two-dimensional nonlinear space-time fractional diffusion model and improve the computational efficiency. First, the second-order backward difference scheme is used to formulate the time approximation, where the time-fractional derivative is approximated by the weighted and shifted Grünwald difference operator. In order to reduce the computation time of the standard FE method, a TGFE algorithm is developed. The specific algorithm is to iteratively solve a nonlinear system on the coarse grid and then to solve a linear system on the fine grid. We prove the scheme stability of the TGFE algorithm and derive a priori error estimate with the convergence result O(Δt 2 + h r + 1 − + H 2r + 2 − 2). Finally, through a two-dimensional numerical calculation, we improve the computational efficiency and reduce the computation time by the TGFE algorithm. KEYWORDS error analysis, nonlinear space-time fractional diffusion model, two-grid finite element algorithm 1 INTRODUCTION Space-time fractional partial differential equations (FPDEs) are a kind of important mathematical models. Because of the complexity of structure for this class of fractional models, people need to develop efficient numerical methods to achieve the numerical solutions. One can see some numerical studies such as difference algorithms for the advection-diffusion equation with space-time fractional

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“…Liu et al [32] developed a numerical scheme based on finite difference formulation and the Fourier spectral method to solve time fractional ACE in one and two space dimensions. Smooth and nonsmooth solutions for nonlinear space fractional ACE were studied by Yin et al [33] using a fast algorithm based on time two-mesh finite difference method. Inc et al [34] reduced the time fractional ACE and time fractional Klein-Gordon equations into the corresponding nonlinear fractional ODEs and employed an explicit power series method to solve these fractional ODEs.…”

Section: Introductionmentioning

confidence: 99%

A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions

et al. 2020

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We present a collocation approach based on redefined cubic B-spline (RCBS) functions and finite difference formulation to study the approximate solution of time fractional Allen-Cahn equation (ACE). We discretize the time fractional derivative of order α ∈ (0, 1] by using finite forward difference formula and bring RCBS functions into action for spatial discretization. We find that the numerical scheme is of order O(h 2 + t 2-α) and unconditionally stable. We test the computational efficiency of the proposed method through some numerical examples subject to homogeneous/nonhomogeneous boundary constraints. The simulation results show a superior agreement with the exact solution as compared to those found in the literature.

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

Cited by 63 publications

References 49 publications

Approximation methods for the distributed order calculus using the convolution quadrature

Approximation methods for the distributed order calculus using the convolution quadrature

Fast calculation based on a spatial two‐grid finite element algorithm for a nonlinear space–time fractional diffusion model

A numerical investigation of Caputo time fractional Allen–Cahn equation using redefined cubic B-spline functions

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