1、剛性振蕩問題論文：剛性振蕩問題 MIRK 方法 A-穩定 L-穩定彌散誤差耗散誤差 P-穩定【中文摘要】剛性振蕩問題是一類兼具剛性和振蕩性兩種特性的 問題,在航空航天、機械學、化學動力學、分子動力學等科技工程領 域中有廣泛應用，研究其高效數值解法有重要的現實意義。剛性和振 蕩性的并存給此類問題的求解帶來了挑戰。 人們一直在努力尋求其高 效可行的數值算法。本文主要是將Mono-implicit Runge-Kutta (MIRK) 方法應用于剛性高振蕩問題的求解，并在盡量保證精度和穩定性的條 件下通過控制方法的彌散誤差及耗散誤差來提高方法的有效性。全文共由五章組成。第一章首先介紹了問題背景、研究

2、動態及前人所取得 的成果,而后闡述了本文的主要工作。第二章主要從經典收斂階的角 度介紹了1-4 級 MIRK 方法簇。第三章對第二章的一些方法簇做了L-穩定性分析，對于某些公式還考慮了代數穩定性。第四章對前面所構 造的滿足一定階條件和穩定性的方法進行了彌散誤差和耗散誤差分 析,并得到了一些具有高階彌散誤差和耗散誤差的公式。特別地,我們得到了兩類三級 P-穩定 MIRK 公式。第五章進行了數值實驗，實驗結 果表明了 MIRK 方法求解剛性振蕩問題的有效性。同時我們通過比較 發現具有高階耗散誤差的方法比具有高階彌散誤差的方法更適合于 此類問題的求解。【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.【關鍵詞】剛性振蕩問題 MIRK 方法 A-穩定 L-穩定 彌散誤差 耗散誤差 P-穩定【英文關鍵詞】Stiff oscillatory problemsMono-implicit Run ge-Kutta methods A-stability L-stability Dispersi onerror Dissipati on errorP-stability【目錄】求解剛性振蕩問題的單調隱式 Runge-Kutta 方法摘要5-6 Abstract 6 第一章 引言 8-10 第二章 問題與方法10-182.1 問題描述 10-132.2 1-4 級單調隱式 Runge-Kutta方法 13-182.2.1 一級單調隱式 Runge-Kutta 方法 132.2.2二級單