A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations

Yin, Baoli; Liu, Yang; Li, Hong

doi:10.1016/j.amc.2019.124799

Cited by 33 publications

(31 citation statements)

References 41 publications

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“…Here, the TGFE method is a suitable choice for the fast computing. In the future, we will solve more complex space–time fractional differential equation models by the current or improved fast TG algorithms combined with other second‐order time approximation formulas [41, 42].…”

Section: Discussionmentioning

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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Section: Discussionmentioning

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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“…Fractional differential equations (FDEs) have been greatly developed, and in fact they can be found in many scientific and engineering fields, and include some practical models, such as fractional diffusion or advection models [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], fractional water wave models [18,19], fractional telegraph models [20], fractional Cable models [21,22], and fractional control models [23]. It is very important to find the solution of FDEs.…”

Section: Introductionmentioning

confidence: 99%

Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

Hou

Cao

et al. 2020

Mathematics

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In this article, some high-order time discrete schemes with an H 1 -Galerkin mixed finite element (MFE) method are studied to numerically solve a nonlinear distributed-order sub-diffusion model. Among the considered techniques, the interpolation approximation combined with second-order σ schemes in time is used to approximate the distributed order derivative. The stability and convergence of the scheme are discussed. Some numerical examples are provided to indicate the feasibility and efficiency of our schemes.

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“…Wang [31] constructed a highorder compact FD scheme to solve the fractional mobile/immobile convection-diffusion equations, and gave a Richardson extrapolation algorithm to improve the temporal convergence accuracy. Yin et al [32] constructed a generalized BDF2-θ scheme for the fractional mobile/immobile transport equations with the initial singularity of the time fractional derivative. However, we find that there is no report about the finite volume element method with the weighted and shifted Grünwald-Letnikov difference (WSGD) formula to solve the nonlinear time fractional mobile/immobile transport equations.…”

Section: Introductionmentioning

confidence: 99%

Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations

Zhao

Fang

et al. 2020

Adv Differ Equ

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In this article, a finite volume element method with the second-order weighted and shifted Grünwald difference (WSGD) formula is proposed and studied for nonlinear time fractional mobile/immobile transport equations on triangular grids. By using the WSGD formula of approximating the Riemann-Liouville fractional derivative and an interpolation operator I * h , a second-order fully discrete finite volume element (FVE) scheme is formulated. The existence, uniqueness, and unconditional stability for the fully discrete FVE scheme are derived, the optimal a priori error estimates in L ∞ (L 2 (Ω)) and L 2 (H 1 (Ω)) norms are obtained, in which the convergence orders are independent of the fractional parameters. At the end of this article, two numerical examples with different nonlinear terms are given to verify the feasibility and effectiveness.

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A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations

Cited by 33 publications

References 41 publications

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

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

Some Second-Order σ Schemes Combined with an H1-Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation

Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations

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