scholarly journals Generation and Analysis of a New Implicit Difference Scheme for the Korteweg-de Vries Equation

2018 ◽  
Vol 173 ◽  
pp. 03006
Author(s):  
Yuri Blinkov ◽  
Vladimir Gerdt ◽  
Konstantin Marinov
Keyword(s):  
Difference Scheme ◽  
Vries Equation ◽  
Implicit Difference Scheme ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
Algorithmic Approach ◽  
De Vries ◽  
Volume Method ◽  
Modified Equation ◽  
Two Parameter

In this paper we apply our computer algebra based algorithmic approach to construct a new finite difference scheme for the two-parameter form of the Korteweg-de Vries equation. The approach combines the finite volume method, numerical integration and difference elimination. Being implicit, the obtained scheme is consistent and unconditionally stable. The modified equation for the scheme shows that its accuracy is of the second order in each of the mesh sizes. Using exact one-soliton solution, we compare the numerical behavior of the scheme with that of the other two schemes known in the literature and having the same order of accuracy. The comparison reveals numerical superiority of our scheme.

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Allaberen Ashyralyev ◽  
Okan Gercek
Keyword(s):  
Difference Scheme ◽  
Parabolic Equations ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Second Order ◽  
Well Posedness ◽  
Parabolic Differential Equations ◽  
Implicit Difference Scheme ◽  
Order Of Accuracy

We are interested in studying a second order of accuracy implicit difference scheme for the solution of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this difference scheme is established. In an application, coercivity estimates in Hölder norms for approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic differential equations are obtained.

scholarly journals An Unconventional Finite Difference Scheme for Modified Korteweg-de Vries Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Canan Koroglu ◽  
Ayhan Aydin
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Truncation Error ◽  
Vries Equation ◽  
Mkdv Equation ◽  
Numerical Results ◽  
De Vries ◽  
Nsfd Scheme ◽  
Theta Method

A numerical solution of the modified Korteweg-de Vries (MKdV) equation is presented by using a nonstandard finite difference (NSFD) scheme with theta method which includes the implicit Euler and a Crank-Nicolson type discretization. Local truncation error of the NSFD scheme and linear stability analysis are discussed. To test the accuracy and efficiency of the method, some numerical examples are given. The numerical results of NSFD scheme are compared with the exact solution and a standard finite difference scheme. The numerical results illustrate that the NSFD scheme is a robust numerical tool for the numerical integration of the MKdV equation.

Integration of the nonlinear modified Korteweg-de Vries equation with a loaded term

2020 ◽  
Vol 2020 (2) ◽  
pp. 85-98
Author(s):  
A.B. Khasanov ◽  
T.J. Allanazarova
Keyword(s):  
Vries Equation ◽  
De Vries ◽  
Loaded Term
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