scholarly journals Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1975
Author(s):  
M. Luísa Morgado ◽  
Magda Rebelo ◽  
Luís L. Ferrás
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Uniform Mesh ◽  
Numerical Schemes ◽  
The Stability ◽  
Distributed Order

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

Finite difference scheme for the time-space fractional diffusion equations

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li
Keyword(s):  
Finite Difference ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Order Accuracy ◽  
Fractional Diffusion Equations ◽  
Time Space ◽  
Liouville Fractional Derivative ◽  
The Stability

AbstractIn this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.

A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

2016 ◽  
Vol 27 (09) ◽  
pp. 1650097 ◽  
Author(s):  
A. H. Encinas ◽  
V. Gayoso-Martínez ◽  
A. Martín del Rey ◽  
J. Martín-Vaquero ◽  
A. Queiruga-Dios
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Higher Order ◽  
Finite Difference Schemes ◽  
Nonlinear Phenomena ◽  
Klein Gordon ◽  
Implicit Finite Difference ◽  
The Stability ◽  
Stability And Consistency ◽  
New Algorithms

In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

scholarly journals Novel Second-Order Accurate Implicit Numerical Methods for the Riesz Space Distributed-Order Advection-Dispersion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
X. Wang ◽  
F. Liu ◽  
X. Chen
Keyword(s):  
Numerical Methods ◽  
Quadrature Rule ◽  
Second Order ◽  
Riesz Space ◽  
Stability And Convergence ◽  
Dispersion Equations ◽  
Advection Dispersion ◽  
Alternating Direction ◽  
The Stability ◽  
Distributed Order

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis.

The Finite Difference Scheme for 3d Mathematical Modeling of a Wood Drying Process

2001 ◽  
Vol 1 (2) ◽  
pp. 125-137 ◽  
Author(s):  
Raimondas Čiegis ◽  
Vadimas Starikovičius
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
The Stability ◽  
Robin Boundary

AbstractThis work discusses issues on the design and analysis of finite difference schemes for 3D modeling the process of moisture motion in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for Robin boundary conditions. The influence of boundary conditions is investigated, and results of numerical experiments are presented.

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.

THE FINITE DIFFERENCE SCHEME FOR MATHEMATICAL MODELING OF WOOD DRYING PROCESS

2001 ◽  
Vol 6 (1) ◽  
pp. 48-57 ◽  
Author(s):  
R. Čiegis ◽  
V. Starikovičius
Keyword(s):  
Mathematical Modeling ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
Moisture Movement ◽  
Different Types ◽  
The Stability

This work discusses issues on the design of finite difference schemes for modeling the moisture movement process in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for different types of boundary conditions.

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