On numerical solution of the time-fractional diffusion-wave equation with the fictitious time integration method

We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples.

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Numerical solution of fractional diffusion-wave equation based on fractional multistep method

Applied Mathematical Modelling ◽

10.1016/j.apm.2013.11.069 ◽

2014 ◽

Vol 38 (14) ◽

pp. 3652-3661 ◽

Cited By ~ 26

Author(s):

J.Y. Yang ◽

J.F. Huang ◽

D.M. Liang ◽

Y.F. Tang

Keyword(s):

Wave Equation ◽

Numerical Solution ◽

Fractional Diffusion ◽

Multistep Method ◽

Diffusion Wave

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An improved higher-order explicit time integration method with momentum corrector for linear and nonlinear dynamics

Applied Mathematical Modelling ◽

10.1016/j.apm.2021.05.013 ◽

2021 ◽

Author(s):

Tianhao Liu ◽

Fanglin Huang ◽

Weibin Wen ◽

Shanyao Deng ◽

Shengyu Duan ◽

...

Keyword(s):

Nonlinear Dynamics ◽

Integration Method ◽

Time Integration ◽

Higher Order ◽

Explicit Time Integration ◽

Time Integration Method ◽

Linear And Nonlinear

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An Improved Time Integration Method Based on Galerkin Weak Form with Controllable Numerical Dissipation

Applied Sciences ◽

10.3390/app11041932 ◽

2021 ◽

Vol 11 (4) ◽

pp. 1932

Author(s):

Weixuan Wang ◽

Qinyan Xing ◽

Qinghao Yang

Keyword(s):

High Frequency ◽

Integration Method ◽

Time Integration ◽

Low Frequency ◽

Weak Form ◽

High Accuracy ◽

Numerical Dissipation ◽

Frequency Range ◽

Time Integration Method ◽

Numerical Damping

Based on the newly proposed generalized Galerkin weak form (GGW) method, a two-step time integration method with controllable numerical dissipation is presented. In the first sub-step, the GGW method is used, and in the second sub-step, a new parameter is introduced by using the idea of a trapezoidal integral. According to the numerical analysis, it can be concluded that this method is unconditionally stable and its numerical damping is controllable with the change in introduced parameters. Compared with the GGW method, this two-step scheme avoids the fast numerical dissipation in a low-frequency range. To highlight the performance of the proposed method, some numerical problems are presented and illustrated which show that this method possesses superior accuracy, stability and efficiency compared with conventional trapezoidal rule, the Wilson method, and the Bathe method. High accuracy in a low-frequency range and controllable numerical dissipation in a high-frequency range are both the merits of the method.

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