Frequency Determination And Step-length Control For A Kind Of Exponentially Fitted Rosenbrock Methods

Regarding the solutions of the initial value problem of ordinary differential equations,domestic and foreign scholars have given many related research results.In several important scientific fields such as quantum mechanics,elasticity and electronics,many problems can be represented by mathematical models of ordinary differential equations,which are usually characterized by oscillation and periodicity.The solutions of the equations show a more ob-vious exponential or triangle form.In these problems,it is difficult to solve them with classic numerical methods accurately.Therefore,just like the frequency and period of the oscilla-tion,we can combine these parameters related to the solutions of the system,and adopt the idea of function fitting,choosing the basis functions as{e^?t,e^-?t}or{sin(?t),cos(?t)}to construct the corresponding fitting methods.There are some mature exponential fitted linear multi-step methods,the exponential fitted Runge-Kutta methods and the exponential fitted Runge-Kutta-Nystr¨om methods,but the research about the exponential fitted Rosen-brock methods is still in its infancy.As the special type of the explicit methods,Rosenbrock methods not only keep the relative stability of the diagonal implicit Runge-Kutta methods,but also save the amount of calculation,and they have a significant computational effect for solving stiff problems.At present,some scholars have combined the exponential fitted with the classic Rosenbrock methods to construct some variable coefficient exponential fit-ted Rosenbrock methods,but the frequency and step size of the methods need to be given in advance,so they are not strictly variable coefficient methods.If the frequency and step size are not selected well,the fitted results will be unsatisfactory.This thesis will perform the exponential fitted for the extended Rosenbrock methods,and combine the obtained method with the embedded Rosenbrock methods to construct the embedded variable coefficient ex-ponential fitted Rosenbrock(3,2)methods.The frequency?through the error analysis can give a specific value before each step of calculation,and use the Richardson extrapolation methods to control the step length to achieve variable step length calculation,so that the constructed method can solve the models of the initial value problem of ordinary differen-tial equations with different parameters more accurately,and the accuracy can be improved by one.In chapter 1,we introduce the background of initial value problems of ordinary differ-ential equations,and give the research status of Rosenbrock methods and exponential fitted method and its relative frequency determination and step control.In Chapter 2,for solving first-order ordinary differential equations,we perform expo-nential fitted through the extended Rosenbrock methods to construct a class of 1-3-stage extended exponential fitted Rosenbrock methods,and give the local truncation error,fre-quency determination and stability analysis of the exponential fitted methods.In Chapter 3,the extended exponential fitted Rosenbrock method and the embedded Rosenbrock method are combined to construct a kind of embedded variable coefficient ex-ponential fitted Rosenbrock(3,2)method,and the frequency determination strategy is given and the adaptive variable step length calculation is realized by using the Richardson extrap-olation method to control the step length.In addition,the construction of the exponential fitted W method is analyzed.In Chapter 4,the validity of the extended 2-3 stage exponential fitted Rosenbrock meth-ods constructed in this thesis can be upgraded,and the exponential fitted Rosenbrock(3,2)method can be improved by one,and the advantages of the constructed adaptive step size method are demonstrated by comparing the number of steps accepted,the number of dis-carded steps and the error of the calculation process with other methods.In the last chapter,we summarize the content of this thesis,and give some reference suggestions and prospects in the field of exponential fitted Rosenbrock methods in the future.