Two Classes Of Functionally Fitted Rosenbrock Methods For Solving Stiff Ordinary Differential Equations

With the deep research on initial value problems related to systems of stiff ordinary differential equations,domestic and foreign scholars have given many effective special RungeKutta methods which include the Diagonal Implicit Runge-Kutta methods and Rosenbrock methods.Functionally fitting methods is a kind of methods that approximates the solution of stiff ordinary differential equations on a local interval.We consider constructing an exponentially fitting function on its solution interval so that it approximates the solution curve of the original equations.An efficient and accurate algorithm is to combine the Runge-Kutta methods with exponentially fitting to solve stiff problems.At the same time,the trigonometrically fitting Runge-Kutta methods are constructed,which exactly integrates differential initial value problems whose solutions are linear combinations of functions of the form ei?x and e-i?x or cos(?x)and sin(?x)(? > 0).The new methods have more advantageous which compared to the traditional methods.Scholars have done much work on the exponentially fitting and trigonometrically fitting Diagonal Implicit Runge-Kutta methods,but the exponentially fitting and trigonometrically fitting Rosenbrock methods have not been used to solve stiff problems,and Rosenbrock methods have a smaller calculation than the Diagonal Implicit Runge-Kutta methods.Therefore,this paper will use the exponentially fitting and trigonometrically fitting Rosenbrock methods to solve initial value problems related to systems of stiff ordinary differential equations.In the first chapter,the background of initial value problems related to systems of stiff ordinary differential equations are introduced,and the development and improvement of the Rosenbrock methods by the early scholars are given.In Chapter 2 and 3,a class of second stage exponentially fitting and trigonometrically fitting Rosenbrock methods of algebraic order 2 are constructed,and the specific methods of fixed coefficients are given respectively,and we proved that there is no such three stage exponentially fitting and trigonometrically fitting Rosenbrock methods of algebraic order 3.Finally,it is verified that the specific methods of fixed coefficients are A-stable.In the fourth chapter,numerical experiments show the efficiency of our new method when it is compared with other methods in the convergence and the calculation time.Finally,we get the conclusion that it is basically consistent with the theory through the experimental results.In the last chapter,we make a summary of this article,and some ideas and plans for the author's later period are given for other models and methods which not covered in this paper.