On Stability of Some Finite Difference Schemes for the Korteweg-de Vries Equation

1975 ◽  
Vol 39 (1) ◽  
pp. 229-236 ◽  
Author(s):  
Katuhiko Goda
Keyword(s):  
Finite Difference ◽  
Vries Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
De Vries

Finite-difference schemes for the korteweg—de vries equation

2000 ◽  
Vol 36 (5) ◽  
pp. 789-797 ◽  
Author(s):  
V. I. Mazhukin ◽  
P. P. Matus ◽  
I. A. Mikhailyuk
Keyword(s):  
Finite Difference ◽  
Vries Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
De Vries

scholarly journals Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

2020 ◽  
Vol 49 ◽  
pp. 57-69
Author(s):  
Vladimir Ivanovich Mazhukin ◽  
◽  
Aleksandr Viktorovich Shapranov ◽  
Elena Nikolaevna Bykovskaya
Keyword(s):  
Finite Difference ◽  
Vries Equation ◽  
Approximation Error ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
The Family ◽  
Implicit Difference Schemes ◽  
Solution Accuracy ◽  
Modified Equation

A family of weighted two-layer finite-difference schemes is presented. Using the example of the numerical solution of model problems on the propagation of a single soliton and the interaction of two solitons, the high quality of explicit-implicit schemes of the Crank-Nichols type with the parameter σ = 0.5 and the order of approximation O(Δt2 + Δx2) isshown. Completely implicit two-layer difference schemes with the parameter σ = 1 and O (Δt+ Δx2) are characterized by absolute stability with a low solution accuracy due to a highapproximation error. The family of explicitly implicit difference schemes is absolutely unstable if the explicitness parameter σ <0.5 prevails. Analysis of the structure of the approximation error, performed using the modified equation method, confirmed the results of numerical simulation.

scholarly journals Error estimates of finite difference schemes for the Korteweg–de Vries equation

2018 ◽  
Vol 40 (1) ◽  
pp. 628-685 ◽  
Author(s):  
Clémentine Courtès ◽  
Frédéric Lagoutière ◽  
Frédéric Rousset
Keyword(s):  
Cauchy Problem ◽  
Finite Difference ◽  
Vries Equation ◽  
Strong Solutions ◽  
Finite Difference Schemes ◽  
Order Convergence ◽  
First Order ◽  
De Vries ◽  
The Cauchy Problem ◽  
Dispersive Term

Abstract This article deals with the numerical analysis of the Cauchy problem for the Korteweg–de Vries equation with a finite difference scheme. We consider the explicit Rusanov scheme for the hyperbolic flux term and a 4-point $\theta $-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant–Friedrichs–Lewy condition when $\theta \geq \frac{1}{2}$ and under an ‘Airy’ Courant–Friedrichs–Lewy condition when $\theta &lt;\frac{1}{2}$. More precisely, we get a first-order convergence rate for strong solutions in the Sobolev space $H^s(\mathbb{R})$, $s \geq 6$ and extend this result to the nonsmooth case for initial data in $H^s(\mathbb{R})$, with $s\geq \frac{3}{4}$, at the price of a reduction in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when $s\geq 3$.

scholarly journals Conservative finite difference schemes for the modified Camassa-Holm equation

JSIAM Letters ◽  
2011 ◽  
Vol 3 (0) ◽  
pp. 37-40 ◽  
Author(s):  
Yuto Miyatake ◽  
Takayasu Matsuo ◽  
Daisuke Furihata
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Holm Equation
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