刚性振荡问题论文：刚性振荡问题MIRK方法A-稳定L-稳定弥散误差耗散误差P-稳定

1、刚性振荡问题论文：刚性振荡问题 MIRK 方法 A-稳定 L-稳定弥散误差耗散误差 P-稳定【中文摘要】刚性振荡问题是一类兼具刚性和振荡性两种特性的 问题,在航空航天、机械学、化学动力学、分子动力学等科技工程领 域中有广泛应用，研究其高效数值解法有重要的现实意义。刚性和振 荡性的并存给此类问题的求解带来了挑战。 人们一直在努力寻求其高 效可行的数值算法。本文主要是将Mono-implicit Runge-Kutta (MIRK) 方法应用于刚性高振荡问题的求解，并在尽量保证精度和稳定性的条 件下通过控制方法的弥散误差及耗散误差来提高方法的有效性。全文共由五章组成。第一章首先介绍了问题背景、研究

2、动态及前人所取得 的成果,而后阐述了本文的主要工作。第二章主要从经典收敛阶的角 度介绍了1-4 级 MIRK 方法簇。第三章对第二章的一些方法簇做了L-稳定性分析，对于某些公式还考虑了代数稳定性。第四章对前面所构 造的满足一定阶条件和稳定性的方法进行了弥散误差和耗散误差分 析,并得到了一些具有高阶弥散误差和耗散误差的公式。特别地,我们得到了两类三级 P-稳定 MIRK 公式。第五章进行了数值实验，实验结 果表明了 MIRK 方法求解刚性振荡问题的有效性。同时我们通过比较 发现具有高阶耗散误差的方法比具有高阶弥散误差的方法更适合于 此类问题的求解。【英文摘要】Stiff oscillatory

3、problems with two characteristics(i.e.,stiff ness and oscillati on) widelyappear invarious ffelds of moder n scie nee and tech no logy, such asaerospace,mecha ni cs, chemical dyn amics, molecular dyn amics etc.The studies in the highly-effcient numerical methods for this kind ofproblems are of impor

4、tant practicalsigniffcance. The coexistenee ofstiffness and oscillationbrings challengeto numer-ical solution of this problem. People have been trying to ffndsome effcie nt an dimpleme ntable nu merical algorithms for theseproblems all the timen this paper we mainly consider using mono-implicitRun g

5、e-Kutta (MIRK)methods to solve stiff high-oscillatory problems. Wemake our methods to be highly-effcie nt through the con troll ing of thedispers ion errors and dissipation errors undercertain given conditions ofaccuracy and stability as much as possible. The full textis composed offfve chapters .In

6、 chapter 1, we in troduce the research backgro und,previous relative resultsa nd the main work of this paper.I n chapter 2, wein troduce MIRK methods of stages 1-4 from the point ofclassical orders ofconvergencen chapter 3, we analyze the L-stability of these methodsobtained in chapter2, and for som

7、eformulas, algebraic stability is alsoconsideredn chapter 4, the dispersion errors and dissipationerrors of these methods area nalyzed un der the obta ined con diti ons ofaccuracy and stability in chapters 2 and3 , moreover, we con struct somemethods with higher-order dispers ion errors an ddissipat

8、io n errors .Inparticular, weobtained two types of P-stable MIRKformulas.ln chapter 5, some numerical experime nts are carried out to show the effcie ncyof some give nmethods for solvi ng stiff oscillatory problems. By comparing ofthenumerical experiments, weffnd that the methods with dissipati on e

9、rrorof higherorder rather tha n the methods with dispers ion error of higherorder are suitable forsolving stiff oscillatory problems.【关键词】刚性振荡问题 MIRK 方法 A-稳定 L-稳定 弥散误差 耗散误差 P-稳定【英文关键词】Stiff oscillatory problemsMono-implicit Run ge-Kutta methods A-stability L-stability Dispersi onerror Dissipati on errorP-stability【目录】求解刚性振荡问题的单调隐式 Runge-Kutta 方法摘要5-6 Abstract 6 第一章 引言 8-10 第二章 问题与方法10-182.1 问题描述 10-132.2 1-4 级单调隐式 Runge-Kutta方法 13-182.2.1 一级单调隐式 Runge-Kutta 方法 132.2.2二级单