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High Order Time Stepping Methods for Nonlinear Evolution Equations

机译：非线性发展方程的高阶时间步进方法

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Many important engineering problems require the time evolution of nonlinear processes. There are many methods in the literature for approximating homogeneous linear evolution equations that are unconditionally stable and take advantage of sparsity structure. This paper demonstrates how to derive high-order time approximation procedures for certain nonlinear evolution equations based on a given high-order approximation procedure for the homogeneous linear problem. For L a fixed differential operator, nonlinear evolution equations of the form u/sub t/ + Lu = J(u) are considered where J(u) is a nonlinear function of u. A stable and high-order algorithm for approximating the linear problem v/sub t/ + Lv = 0 is assumed to be given. This paper shows how to use the approximation algorithm to get high-order approximation for the evaluation equation. The results of computational experiments are given which illustrate the benefits of the time stepping methods. The Korteweg-de Vries equation and a nonlinear Schroedinger equation are considered. The spatial discretization is given by the spectral method with trigonometric polynomials (Fourier method). 2 figures, 1 table. (ERA citation 05:023383)

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