On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

scholarly journals Conservative Linear Difference Scheme for Rosenau-KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinsong Hu ◽  
Youcai Xu ◽  
Bing Hu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Difference ◽  
Second Order Convergence ◽  
Initial Boundary Value ◽  
Theoretical Results

A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed. The difference scheme simulates two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

scholarly journals An Average Linear Difference Scheme for the Generalized Rosenau-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Maobo Zheng ◽  
Jun Zhou
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Discrete Energy ◽  
The Difference ◽  
Initial Boundary Value ◽  
Theoretical Results

An average linear finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-KdV equation is proposed. The existence, uniqueness, and conservation for energy of the difference solution are proved by the discrete energy norm method. It is shown that the finite difference scheme is 2nd-order convergent and unconditionally stable. Numerical experiments verify that the theoretical results are right and the numerical method is efficient and reliable.

scholarly journals Conservative Difference Scheme for Generalized Rosenau-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yan Luo ◽  
Youcai Xu ◽  
Minfu Feng
Keyword(s):  
Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Discrete Analogue ◽  
The Difference ◽  
Second Order Convergence ◽  
Initial Boundary Value ◽  
Conservative Difference ◽  
Theoretical Results

A conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of generalized Rosenau-KdV equation is proposed. The difference scheme shows a discrete analogue of the main conservation law associated to the equation. On the other hand the scheme is implicit and stable with second order convergence. Numerical experiments verify the theoretical results.

scholarly journals Numerical Solution of Stochastic Hyperbolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Necmettin Aggez ◽  
Maral Ashyralyyewa
Keyword(s):  
Difference Scheme ◽  
Numerical Solution ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergence Estimate ◽  
Numerical Examples ◽  
Convergence Estimates ◽  
The Difference ◽  
Initial Boundary Value

A two-step difference scheme for the numerical solution of the initial-boundary value problem for stochastic hyperbolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of the difference scheme are obtained for different initialboundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

scholarly journals On the stable difference scheme for the time delay telegraph equation

2020 ◽  
Vol 99 (3) ◽  
pp. 105-119
Author(s):  
A. Ashyralyev ◽  
◽  
K. Turk ◽  
D. Agirseven ◽  
◽  
...  
Keyword(s):  
Time Delay ◽  
Difference Scheme ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Telegraph Equation ◽  
Stability Estimates ◽  
The Difference ◽  
Initial Boundary Value

The stable difference scheme for the approximate solution of the initial boundary value problem for the telegraph equation with time delay in a Hilbert space is presented. The main theorem on stability of the difference scheme is established. In applications, stability estimates for the solution of difference schemes for the two type of the time delay telegraph equations are obtained. As a test problem, one-dimensional delay telegraph equation with nonlocal boundary conditions is considered. Numerical results are provided.

scholarly journals On the evolution of solutions of Burgers equation on the positive quarter-plane

2019 ◽  
Vol 52 (1) ◽  
pp. 237-248
Author(s):  
Esen Hanaç
Keyword(s):  
Boundary Condition ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Condition ◽  
Quarter Plane ◽  
Initial Boundary Value ◽  
Good Agreement

AbstractIn this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; $\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

scholarly journals Kawahara-Burgers Equation on a Strip

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
N. A. Larkin
Keyword(s):  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regular Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Initial Boundary Value ◽  
Channel Type

An initial-boundary value problem for the 2D Kawahara-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in theL2-norm.

The Comparison for Several Difference Methods in Heat Conduction Equations

2011 ◽  
Vol 282-283 ◽  
pp. 399-402
Author(s):  
Fan Lei Meng
Keyword(s):  
Heat Conduction ◽  
Difference Scheme ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Accuracy ◽  
Third Order ◽  
Numerical Experimentation ◽  
Valuable Method ◽  
Accuracy Difference ◽  
Initial Boundary Value

In this paper, one-dimensional heat conduction equations is studied, many difference Schemes have been proposed to solve it. In order to find a high accuracy difference scheme in all the methods, we give a numerical experimentation in this paper. by numerical experimentation, a high accuracy difference scheme for solving Heat conduction equations initial boundary value problem is found, according to the truncation error and stability analysis ,we find its accuracy is better-then- third-order in time and space direction. this is a valuable method and better then the others this is a high accuracy difference Scheme. this scheme is a valuable method in Heat conduction and Fluid mechanics.

scholarly journals Some features of numerical diagnostics of instantaneous blow-up of the solution by the example of solving the equation of slow diffusion

2021 ◽  
pp. 77-86
Author(s):  
И.В. Пригорный ◽  
А.А. Панин ◽  
Д.В. Лукьяненко
Keyword(s):  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Ill Posed ◽  
The Difference ◽  
Initial Boundary Value ◽  
Richardson Extrapolation Method ◽  
Posteriori Estimation

В работе демонстрируется, как метод апостериорной оценки порядка точности разностной схемы по Ричардсону позволяет сделать вывод о некорректности постановки (в смысле отсутствия решения) решаемой численно начально-краевой задачи для уравнения в частных производных. Это актуально в ситуации, когда аналитическое доказательство некорректности постановки ещё не получено или принципиально невозможно. The paper demonstrates how the method of a posteriori estimation of the order of accuracy for the difference scheme according to the Richardson extrapolation method allows one to conclude that the formulation of the numerically solved initial-boundary value problem for a partial differential equation is ill-posed (in the sense of the absence of a solution). This is important in a situation when the ill-posedness of the formulation is not analytically proved yet or cannot be proved in principle.

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