Exponentially Fitted Rosenbrock Methods With Variable Coefficients For Solving First-order Ordinary Differential Equations

In the fields of chemical reactions,thermonuclear reactions,aeronautical engineering,and other important engineering studies,the solutions of many mathematical models are periodicity,oscillation or obvious exponential form.Systems with solutions of this nature are typically modeled by stiff ordinary differential equations.The solutions of these stiff systems are difficult to solve analytically and require numerical simulation using suitable numerical methods.At the same time,certain parameters that are closely related to the properties of the solution,such as the frequency and period of the oscillation,can some-times be estimated in advance based on actual problems.Therefore,it is necessary to make reasonable use of these parameters to construct a new approah that is suitable for solving models with this type of solution.Function fitting methods make full use of these parame-ters,and can integrate exactly a set of suitable basis functions which are selected according to the form of solution.The construction of exponential fitting methods,that means the basis functions of the function fitting methods are{e^?t,e^-?t},??C,which can better approach the solutions of the initial value problems with oscillatory or exponential solutions.The existing research results mainly focus on the exponential fitting Runge-Kutta methods?the exponential fitting multi-step methods and the exponential fitting block methods,etc.It is rare to see the exponential fitting Rosenbrock methods.As explicit methods,Rosenbrock methods have the characteristics of easy implementation and low computational complexity,and retains the good stability of Diagonal Implicit Runge-Kutta methods.It is suitable for solving stiff differential equations.At present,some scholars have combined the exponen-tial fitting methods with Rosenbrock methods to construct exponential fitting Rosenbrock methods with fixed coefficients,but the properties of the solution are not rationally utilized.In this paper,we will make full use of the parameters that characterize the solution of the differential equation systems,and construct the exponential fitting Rosenbrock method-s with variable coefficients,so that the method can solve the differential equation models with different parameters in a targeted manner.In the first chapter,the background of initial value problems of ordinary differential equations are introduced,and the research status of function fitting methods and Rosenbrock methods are given.In chapter 2,we give the definition of the function fitting Rosenbrock methods as well as the existence and uniqueness theorem,and study the coefficients of function fitting Rosen-brock methods are independent of t for a set of separable basis functions.In chapter 3,the basis function of function fitting Rosenbrock methods are set as{e^?t,e^-?t},??C,and we obtaine exponential fitting conditions for Rosenbrock meth-ods.Then,combined with the order conditions,a class of second satage exponentially fitting Rosenbrock methods with variable coefficients of second-order and a class of three satage exponentially fitting Rosenbrock methods with variable coefficients of three-order are contructed.After that,the linear stability of exponentially fitted Rosenbrock methods with variable coefficients is analyzed.In chapter 4,numerical experiments are accompanied to show the efficiency and advan-tages of the exponentially fitted Rosenbrock methods with variable coefficients compared with other three methods of calculation time and precision.In the last chapter,we make a summary of the work of this text,and some referential research directions are given combined with the content of this paper.