ABSTRACT

A form of Rosenbrock-type methods optimal in terms of the number of non-zero parameters and computational costs per step is considered. A technique of obtaining \((m, k)\)-methods from some well-known Rosenbrock-type methods is justified. Formulas for transforming the parameters of \((m,k)\)-schemes and for obtaining a stability function are given for two canonical representations of the schemes. An \(L\)-stable \((3, 2)\)-method of order 3 is proposed, which requires two evaluations of the function: one evaluation of the Jacobian matrix and one \(LU\)-decomposition per step. A variable step size integration algorithm based on the \((3,2)\)-method is formulated. It provides a numerical solution for both explicit and implicit systems of ODEs. Numerical results are presented to show the efficiency of the new algorithm.

中文翻译：

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（m，k）-型求解微分代数和刚性系统的方案

摘要

考虑了一种在非零参数数量和每步计算成本方面最优的Rosenbrock型方法。从某些著名的Rosenbrock型方法中获得\（（m，k）\）方法的技术是合理的。针对该方案的两种规范表示，给出了用于转换\（（m，k）\）-方案的参数并获得稳定性函数的公式。提出了阶数为3的\（L \） -稳定 \（（3，2）\） -方法，该函数需要对函数进行两次评估：一次对Ja​​cobian矩阵进行评估，一次 对每一步进行（（LU \））分解。基于\（（3,2）\）的可变步长积分算法 -制定方法。它为ODE的显式和隐式系统提供了数值解决方案。数值结果表明了该算法的有效性。