数学
索波列夫空间
初值问题
收敛速度
订单(交换)
趋同(经济学)
Korteweg–de Vries方程
空格(标点符号)
数学分析
数学物理
应用数学
物理
量子力学
经济增长
电气工程
频道(广播)
工程类
哲学
语言学
非线性系统
经济
财务
作者
Clémentine Courtès,Frédéric Lagoutìère,Frédéric Rousset
出处
期刊:Ima Journal of Numerical Analysis
日期:2018-12-05
卷期号:40 (1): 628-685
被引量:16
标识
DOI:10.1093/imanum/dry082
摘要
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 $\theta <\frac{1}{2}$. 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 $s\geq 3$.
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