Rosenbrock Methods For Stiff Oscillatory Problems

Sti? oscillatory problems appear in many areas of modern science and tech-nology, and are of the dual characters of sti?ness and oscillation. The studies inthe highly-e?cient numerical method for this kind of problems are of importanttheoretical and practical meanings, but also with some di?culties.Runge-Kutta(RK) methods are a major class of single-step methods for solvingsti? problems. More e?ective special classes of RK methods include Singly ImplicitRunge-Kutta(SIRK) methods, Diagonally Implicit RK (DIRK) methods and Rosen-brock methods and so on. Recently, DIRK methods (especially SDIRK methods)for solving sti? oscillation problems has been widely used. But Rosenbrock methodshave not been used to solve sti? oscillation problems yet, this article will focus onthis area.In chapter 2, we construct two-stage Rosenbrock methods of order two andorder four, three-stage Rosenbrock methods of order three, and four-stage Rosen-brock methods of order five. In chapter 3, we discuss the stability properties ofthese methods, including A-stability, P-stability and L-stability. In chapter 4, westudy the order conditions of the dispersion errors, the dissipation errors and thee-consistent errors of two-stage to four-stage Rosenbrock methods. Then, we con-struct several specific Rosenbrock methods, where the free parameters are chosenin order to achieve the highest dispersion and dissipation orders. In chapter 5, wedo some numerical tests. The given methods are applied to solve some linear andnonlinear di?erential equations with sti? and oscillatory solutions, and comparedwith the SDIRK methods constructed by Franco, G′omez, R′andez(1997).