Diagonally-Implicit Runge-Kutta Methods And Its Exponential Fitting

In this paper,we prove that the order of A-stable three-stage diagonally-implicit Runge-Kutta methods of stage order greater than two is no more than three. An one-parameter family of two-stage diagonally-implicit Runge-Kutta methods of order greater than two and stage order greater than two and a two-parameter family of three-stage diagonally-implicit Runge-Kutta methods of order greater than three and stage order greater than two are constructed, and these families have an explicit stage. Also we discuss the one-point exponential fitting of three-stage diagonally-implicit Runge-Kutta methods with an explicit stage and construct the corresponding A-stable exponentially-fitting Runge-Kutta methods. And we point out that the order of the exponentially-fitting methods will be increased by one unit if the frequency is chosen to be optimal when we apply them to solve numerically a concrete problem. The exponentially-fitting methods newly constructed are applied to solve the problems which have oscillating solutions and stiff differential equations of fast and slow decaying processes and but the computational efficiency in the fast decaying processes can be greatly promoted. Numerical experiments further show the obtained theoretical results, and a significant improvement in efficiency by using exponentially fitting methods.