Numerical Analysis Of Exponential ROSENBROCK Methods

Recently, various kinds of semi-linear problems have emerged in many scientific fields such as mathematics, physics, chemistry, biology, ecology and economy etc. It is of great scientific significance for us to research these semi-linear equations.Mainly based on stability analysis, this thesis comprises of three parts according to different equations and methods. Firstly, we are concerned with exponential additive Runge-Kutta methods when they are applied to equations with one dimensional linear coefficients. Secondly, we study the same method while the equations with high-dimensional linear coefficients. Finally, we focus on exponential Rosenbrock methods when they are applied to the same equations.The thesis is organized as follow.In chapter one, we introduce the background of the study problem of this thesis, present many applications of semi-linear differential equations for the recent years, review some classical methods of the past half century, especially the development of Runge-Kutta methods, Rosenbrock methods and exponential integrators combining these classical methods.In chapter two, we start from additive Runge-Kutta methods, establish exponential additive Runge-Kutta methods and define EB -stability and exponential algebraic stability properties. Then according to the dimensions of the linear coefficients of the equations, we discuss these stability properties of exponential additive Runge-Kutta methods in explicit and implicit situations.In chapter three, with the definitions and conclusions in chapter two, we analyze the order conditions and stability properties of exponential Rosenbrock methods when they are applied to equations with high-dimensional linear coefficients. In the ending part, we concludes studies above and show interested readers the details of researching in the future.