Mono-implicit Runge-Kutta Methods For Solving Stiff Oscillatory Problems

Stiff oscillatory problems with two characteristics(i.e., stiffness and oscillation)widely appear in various ffelds of modern science and technology, such as aerospace,mechanics, chemical dynamics, molecular dynamics etc. The studies in the highly-effcient numerical methods for this kind of problems are of important practicalsigniffcance. The coexistence of stiffness and oscillation brings challenge to numer-ical solution of this problem. People have been trying to ffnd some effcient andimplementable numerical algorithms for these problems all the time.In this paper we mainly consider using mono-implicit Runge-Kutta (MIRK)methods to solve stiff high-oscillatory problems. We make our methods to be highly-effcient through the controlling of the dispersion errors and dissipation errors undercertain given conditions of accuracy and stability as much as possible. The full textis composed of ffve chapters.In chapter 1, we introduce the research background, previous relative resultsand the main work of this paper.In chapter 2, we introduce MIRK methods of stages 1-4 from the point ofclassical orders of convergence.In chapter 3, we analyze the L-stability of these methods obtained in chapter2, and for some formulas, algebraic stability is also considered.In chapter 4, the dispersion errors and dissipation errors of these methods areanalyzed under the obtained conditions of accuracy and stability in chapters 2 and3 , moreover, we construct some methods with higher-order dispersion errors anddissipation errors. In particular, we obtained two types of P-stable MIRK formulas.In chapter 5, some numerical experiments are carried out to show the effciencyof some given methods for solving stiff oscillatory problems. By comparing of thenumerical experiments, we ffnd that the methods with dissipation error of higherorder rather than the methods with dispersion error of higher order are suitable forsolving stiff oscillatory problems.