动态系统仿真中的多种算法

Václav Šátek, J. Kunovsky, J. Petrek
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引用次数: 3

摘要

一种基于泰勒级数的求解常微分方程组的非常有趣和有前途的数值方法出现了。许多实际实验揭示了泰勒级数的潜力,并找到了一种检测和求解大型常微分方程组的方法。一般来说,一个刚性系统包含几个组成部分,其中一些被严重抑制,而其余的几乎保持不变。这一特点迫使所使用的方法选择极小的积分步长,计算的进度可能变得非常缓慢。求解ODE的刚性系统有许多(隐式)方法,从最简单的隐式欧拉法到更复杂的(隐式龙格-库塔法),最后是一般的线性方法。通常每一步都要求解一个相当复杂的辅助方程组。这些事实导致在计算的每一步都要做大量的工作。这就是为什么在使用刚性求解器之前必须三思而后行,并在刚性和非刚性求解器之间做出决定的原因。
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Multiple Arithmetic in Dynamic System Simulation
A very interesting and promising numerical method of solving systems of ordinary differential equations based on Taylor series has appeared. The potential of the Taylor series has been exposed by many practical experiments and a way of detection and solution of large systems of ordinary differential equations has been found. Generally speaking, a stiff system contains several components, some of them are heavily suppressed while the rest remain almost unchanged. This feature forces the used method to choose an extremely small integration step and the progress of the computation may become very slow. There are many (implicit) methods for solving stiff systems of ODE’s, from the most simple such as implicit Euler method to more sophisticated (implicit Runge-Kutta methods) and finally the general linear methods. Usually a quite complicated auxiliary system of equations has to be solved in each step. These facts lead to immense amount of work to be done in each step of the computation. These are the reasons why one has to think twice before using the stiff solver and to decide between the stiff and non-stiff solver.
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