scholarly journals The Compact Finite Difference Method of Two-Dimensional Cattaneo Model

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yating Huang ◽  
Zhe Yin
Keyword(s):  
Finite Difference ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
True Solution ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
The Stability ◽  
Convergence Orders ◽  
Convergence Of The Scheme

In this paper, we propose and analyze the compact finite difference scheme of the two-dimensional Cattaneo model. The stability and convergence of the scheme are proved by the energy method, the convergence orders are 2 in time and 4 in space. We also use the variables separation method to find the true solution of the problem. On this basis, the validity and accuracy of the scheme are verified by numerical experiments.

A predictor–corrector compact finite difference scheme for a nonlinear partial integro-differential equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shufang Hu ◽  
Wenlin Qiu ◽  
Hongbin Chen
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Integro Differential Equation ◽  
Compact Finite Difference Method ◽  
Backward Euler ◽  
Predictor Corrector

Abstract A predictor–corrector compact finite difference scheme is proposed for a nonlinear partial integro-differential equation. In our method, the time direction is approximated by backward Euler scheme and the Riemann–Liouville (R–L) fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, a two-step predictor–corrector (P–C) algorithm called MacCormack method is used. A fully discrete scheme is constructed with space discretization by compact finite difference method. Numerical experiment presents the scheme is in good agreement with the theoretical analysis.

scholarly journals A Dufort-Frankel Difference Scheme for Two-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Zongqi Liang ◽  
Yubin Yan ◽  
Guorong Cai
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
The Stability ◽  
Predictor Corrector ◽  
Theoretical Results ◽  
Methods Numerical

A standard Crank-Nicolson finite-difference scheme and a Dufort-Frankel finite-difference scheme are introduced to solve two-dimensional damped and undamped sine-Gordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictor-corrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

scholarly journals A Fast Compact Finite Difference Method for Fractional Cattaneo Equation Based on Caputo–Fabrizio Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Haili Qiao ◽  
Zhengguang Liu ◽  
Aijie Cheng
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Computational Cost ◽  
Evaluation Procedure ◽  
Spatial Accuracy ◽  
Fast Evaluation ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
Theoretical Results

The Cattaneo equations with Caputo–Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank–Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of OMN2 and require memory of OMN, with M and N the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo–Fabrizio fractional derivative, by which the computational cost is reduced to OMN operations; meanwhile, only OM memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure.

scholarly journals Application of Compact Finite Difference Method for Solving Some Type of Fractional Derivative Equations

2021 ◽  
Vol 15 ◽  
pp. 1324-1335
Author(s):  
Mahboubeh Molavi-Arabshahi ◽  
Zahra Saeidi
Keyword(s):  
Finite Difference ◽  
Fractional Derivative ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Equilibrium Points ◽  
Accurate Solution ◽  
Order System ◽  
Compact Finite Difference Scheme ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method

In this paper, the compact finite difference scheme as unconditionally stable method is applied to some type of fractional derivative equation. We intend to solve with this scheme two kinds of a fractional derivative, first a fractional order system of Granwald-Letnikov type 1 for influenza and second fractional reaction sub diffusion equation. Also, we analyzed the stability of equilibrium points of this system. The convergence of the compact finite difference scheme in norm 2 are proved. Finally, various cases are used to test the numerical method. In comparison to other existing numerical methods, our results show that the scheme yields an accurate solution that is quick to compute.

scholarly journals On high-order compact schemes for the multidimensional time-fractional Schrödinger equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Rena Eskar ◽  
Xinlong Feng ◽  
Ehmet Kasim
Keyword(s):  
Finite Difference ◽  
High Order ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Step Size ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
Fractional Schrödinger Equations ◽  
The One

Abstract In this article, some high-order compact finite difference schemes are presented and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations. The time Caputo fractional derivative is evaluated by the L1 and L1-2 approximation. The space discretization is based on the fourth-order compact finite difference method. For the one-dimensional problem, the rates of the presented schemes are of order $O(\tau ^{2-\alpha }+h^{4})$ O ( τ 2 − α + h 4 ) and $O(\tau ^{3-\alpha }+h^{4})$ O ( τ 3 − α + h 4 ) , respectively, with the temporal step size τ and the spatial step size h, and $\alpha \in (0,1)$ α ∈ ( 0 , 1 ) . For the two-dimensional problem, the high-order compact alternating direction implicit method is used. Moreover, unconditional stability of the proposed schemes is discussed by using the Fourier analysis method. Numerical tests are performed to support the theoretical results, and these show the accuracy and efficiency of the proposed schemes.

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