Research On Obrechkoff Numerical Methods For Oscillatory And Stiff Differential Equations

Because of the importance of ordinary differential equations and their wide application in diffirent fields, through the whole 20th century, the research on numerical solutions of ODEs made enormous progress. Particularly, with the fast development of electronic computer and some famous mathematics softwares, more and more new ideas and approaches which surpass classic methods could be realized. At the same time, the field of numerical methods has been expanded.The computational function of computer is not merely to get numerical results for physical research, the more important thing is, and it has offered this new research means of "computer simulation experiment" to physicist. With the powerful tool of electronic computation, we cast sight on numerical solutions to some comparatively complicated differential equations in the physics field (nonlinear Duffing equation, periodic oscillatory equation and stiff equation) and the research on the corresponding numerical methods.In the physics field, we often encounter some differential equation with wide application, for example, Schr6dinger equation, nonlinear Duffing equation, orbital equation and stiff equation. Most of these equations are first-order or second-order equations with simple form, but seldom of could be analytically solved. Because of low accurate numerical methods and the specific property of the equations, it is hard to get ideal results even in numerical way. There are two representative problems: periodic oscillatory problem and stiff problem.In this desertation, we focus mainly on numerical methods for oscillatory and stiff problems.For the oscillatory problem, we focus on second-order differential equationCommonly, there exist some periodic functions like cos(ωx), sin(ωx) or e^tωx in the approximately analytic solution to periodic oscillatory problem. Due to its oscillatory attribute, it is difficult to solve the problem numerically, and even if it is done, the numerical results often become unstable or even diffused.