Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation
Conservative, high-order numerical schemes for the generalized Korteweg—de Vries equation
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DOI：
10.1098/rsta.1995.0027
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发表时间：
1995-04
期刊：
影响因子：
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通讯作者：
J. Bona;V. Dougalis;O. Karakashian;W. R. McKinney;F. Smith
中科院分区：
文献类型：
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作者：
J. Bona;V. Dougalis;O. Karakashian;W. R. McKinney;F. Smith
分析、实现和检验了一类广义Korteweg-de弗里斯方程周期初值问题解的数值模拟的全离散格式。这些计划可能有任意高阶的空间和时间变量，但在同一时间，他们的功能弱的理论稳定性限制。空间离散化的影响，使用光滑样条的二次或更高的程度，而时间离散化是一个多阶段，隐式，龙格库塔法。证明了数值逼近收敛到初值问题的真解的极限消失的空间和时间离散。此外，我们的计划的特定版本的效率进行了仔细的分析。由此收集的信息是用于调查的孤立波解的不稳定性的某一类这些方程。
A class of fully discrete schemes for the numerical simulation of solutions of the periodic initial-value problem for a class of generalized Korteweg-de Vries equations is analysed, implemented and tested. These schemes may have arbitrarily high order in both the spatial and the temporal variable, but at the same time they feature weak theoretical stability limitations. The spatial discretization is effected using smooth splines of quadratic or higher degree, while the temporal discretization is a multi-stage, implicit, Runge-Kutta method. A proof is presented showing convergence of the numerical approximations to the true solution of the initial-value problem in the limit of vanishing spatial and temporal discretization. In addition, a careful analysis of the efficiency of particular versions of our schemes is given. The information thus gleaned is used in the investigation of the instability of the solitary-wave solutions of a certain class of these equations.