Finite difference method for two-dimensional nonlinear time-fractional subdiffusion equation

2018 ◽  
Vol 21 (4) ◽  
pp. 1046-1072 ◽  
Author(s):  
Changpin Li ◽  
Qian Yi
Keyword(s):  
Explicit Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Smooth Solutions ◽  
Present Scheme ◽  
Fast Solver ◽  
Fractional Differential ◽  
Subdiffusion Equation ◽  
The Stability ◽  
Correction Terms

Abstract In this article, we propose an implicit-explicit scheme combining with the fast solver in space to solve two-dimensional nonlinear time-fractional subdiffusion equation. The applications of implicit-explicit scheme and fast solver will smartly enhance the computational efficiency. Due to the non-smoothness (or low regularities) of solutions to fractional differential equations, correction terms are introduced in the proposed scheme to improve the accuracy of error. The stability and convergence of the present scheme are also investigated. Numerical examples are carried out to demonstrate the efficiency and applicability of the derived scheme for both linear and nonlinear fractional subdiffusion equations with non-smooth solutions.

Enhanced Explicit Scheme to Solve Transient Heat Conduction Problem

2007 ◽  
Vol 2 (1) ◽  
Author(s):  
Ganesh Hegde ◽  
Madhu Gattumane
Keyword(s):  
Heat Conduction ◽  
Correction Factor ◽  
Truncation Error ◽  
Transient Heat Conduction ◽  
Explicit Scheme ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Time Step ◽  
One Dimensional ◽  
Partial Derivatives

Improvement in accuracy without sacrificing stability and convergence of the solution to unsteady diffusion heat transfer problems by computational method of enhanced explicit scheme (EES), has been achieved and demonstrated, through transient one dimensional and two dimensional heat conduction. The truncation error induced in the explicit scheme using finite difference technique is eliminated by optimization of partial derivatives in the Taylor series expansion, by application of interface theory developed by the authors. This theory, in its simple terms gives the optimum values to the decision vectors in a redundant linear equation. The time derivatives and the spatial partial derivatives in the transient heat conduction, take the values depending on the time step chosen and grid size assumed. The time correction factor and the space correction factor defined by step sizes govern the accuracy, stability and convergence of EES. The comparison of the results of EES with analytical results, show decreased error as compared to the result of explicit scheme. The paper has an objective of reducing error in the explicit scheme by elimination of truncation error introduced by neglecting the higher order terms in the expansion of the governing function. As the pilot examples of the exercise, the implementation is aimed at solving one-dimensional and two-dimensional problems of transient heat conduction and compared with the results cited in the referred literature.

scholarly journals High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

2014 ◽  
Vol 4 (3) ◽  
pp. 222-241 ◽  
Author(s):  
Seakweng Vong ◽  
Zhibo Wang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Differential Equation ◽  
Neumann Boundary ◽  
High Order ◽  
Neumann Boundary Conditions ◽  
Stability And Convergence ◽  
Alternating Direction ◽  
Fractional Differential ◽  
The Stability

AbstractA compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.

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.

Chebyshev-Legendre Spectral Domain Decomposition Method for Two-Dimensional Vorticity Equations

2016 ◽  
Vol 19 (5) ◽  
pp. 1221-1241 ◽  
Author(s):  
Hua Wu ◽  
Jiajia Pan ◽  
Haichuan Zheng
Keyword(s):  
Domain Decomposition Method ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
One Dimensional ◽  
Collocation Points ◽  
The Matrix ◽  
Vorticity Equations ◽  
Legendre Spectral Method ◽  
Legendre Galerkin Method ◽  
The Stability

AbstractWe extend the Chebyshev-Legendre spectral method to multi-domain case for solving the two-dimensional vorticity equations. The schemes are formulated in Legendre-Galerkin method while the nonlinear term is collocated at Chebyshev-Gauss collocation points. We introduce proper basis functions in order that the matrix of algebraic system is sparse. The algorithm can be implemented efficiently and in parallel way. The numerical analysis results in the case of one-dimensional multi-domain are generalized to two-dimensional case. The stability and convergence of the method are proved. Numerical results are given.

scholarly journals Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1260
Author(s):  
Elsayed I. Mahmoud ◽  
Viktor N. Orlov
Keyword(s):  
Brownian Particle ◽  
Planck Equation ◽  
Variable Coefficients ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Time Space ◽  
Liouville Derivative ◽  
The Stability ◽  
Fokker Planck

This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative and the Riemann–Liouville derivative are considered in the temporal and spatial directions, respectively. The Riemann–Liouville derivative is approximated by the standard Grünwald approximation and the shifted Grünwald approximation. The stability and convergence of the numerical scheme are discussed. Finally, we provide a numerical example to test the theoretical analysis.

scholarly journals Factorized difference scheme for two-dimensional subdiffusion equation in nonhomogeneous media

2016 ◽  
Vol 99 (113) ◽  
pp. 1-13 ◽  
Author(s):  
Aleksandra Delic ◽  
Sandra Hodzic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Nonhomogeneous Media ◽  
Subdiffusion Equation ◽  
Initial Boundary Value

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional subdiffusion equation in nonhomogeneous media is proposed. Its stability and convergence are investigated. The corresponding error bounds are obtained.

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.

scholarly journals A new compact alternating direction implicit method for solving two dimensional time fractional diffusion equation with Caputo-Fabrizio derivative

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3609-3626
Author(s):  
Mehran Taghipour ◽  
Hossein Aminikhah
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
Theoretical Considerations

In this paper, a new compact alternating direction implicit (ADI) difference scheme is proposed for the solution of two dimensional time fractional diffusion equation. Theoretical considerations are discussed. We show that the proposed method is fourth order accurate in space and two order accurate in time. The stability and convergence of the compact ADI method are presented by the Fourier analysis method. Numerical examples confirm the theoretical results and high accuracy of the proposed scheme.

scholarly journals An Efficient Explicit Decoupled Group Method for Solving Two–Dimensional Fractional Burgers’ Equation and Its Convergence Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
N. Abdi ◽  
H. Aminikhah ◽  
A. H. Refahi Sheikhani ◽  
J. Alavi ◽  
M. Taghipour
Keyword(s):  
Numerical Experiments ◽  
Taylor Expansion ◽  
Truncation Error ◽  
Burgers Equation ◽  
Stability And Convergence ◽  
Group Method ◽  
Two Dimensional ◽  
Cpu Time ◽  
The Stability ◽  
Crank Nicolson

In this paper, the Crank–Nicolson (CN) and rotated four-point fractional explicit decoupled group (EDG) methods are introduced to solve the two-dimensional time–fractional Burgers’ equation. The EDG method is derived by the Taylor expansion and 45° rotation of the Crank–Nicolson method around the x and y axes. The local truncation error of CN and EDG is presented. Also, the stability and convergence of the proposed methods are proved. Some numerical experiments are performed to show the efficiency of the presented methods in terms of accuracy and CPU time.

scholarly journals Efficient and Stable Numerical Methods for Multi-Term Time Fractional Sub-Diffusion Equations

2014 ◽  
Vol 4 (3) ◽  
pp. 242-266 ◽  
Author(s):  
Jincheng Ren ◽  
Zhi-zhong Sun
Keyword(s):  
Fractional Derivatives ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Caputo Fractional Derivatives ◽  
Numerical Examples ◽  
One Dimensional ◽  
Compact Difference ◽  
Spatial Discretisation ◽  
The Stability

AbstractSome efficient numerical schemes are proposed for solving one-dimensional (1D) and two-dimensional (2D) multi-term time fractional sub-diffusion equations, combining the compact difference approach for the spatial discretisation and L1 approximation for the multi-term time Caputo fractional derivatives. The stability and convergence of these difference schemes are theoretically established. Several numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

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