Numerical Analysis Of Rational Exponential Runge-kutta Metheods

The main research of this paper was the numerical methods for solving semi-lineardiferential equations.Firstly, the research background and the research significance was discussed, thehistory of exponential integral factors methods for solving semi-linear diferential equa-tions and the history of algorithm for the computation of matrix exponential also werereviewed. Through the review of the general exponential Runge-Kutta methods of collo-cation type, and the analysis of the three error sources of general exponential Runge-Kuttamethods, we established the rational exponential Runge-Kutta methods.Secondly, the convergence theorem of explicit rational exponential Runge-Kuttamethods was proposed and proved. When need to solve the semi-linear diferential equa-tion with contractive, the determinist function and determinist theorem were establishedfor rational exponential Runge-Kutta methods. For general2-stage2-order,3-stage3-order,5-stage4-order, explicit rational exponential Runge-Kutta methods, the stabilityfunction was raised and visualized them by Python2.7.6, respectively. And we proved thetheorem that the stability region of explicit rational exponential Runge-Kutta methods isnon-empty.Thirdly, we verified the contractive and the order of explicit rational exponentialRunge-Kutta methods by numerical experiments. Two spectral preprocess were giv-en for semi-linear diferential equation, one is the Fourier spectral method, another isthe Chebyshev spectral method. For semi-linear partial diferential equation, such as,Schro¨dinger equation, one-dimensional and two-dimensional Allen-Cahn equation, Kd-V equation, Burgers equation, Kuramoto-Sivashinsky equation, Sine-Gordon equation,were computed numerically by the explicit rational exponential Runge-Kutta methods.