Construction of high order diagonally implicit multistage integration methods for ordinary differential equations

The identification of high order diagonally implicit multistage integration methods with appropriate stability properties requires the solution of high dimensional nonlinear equation systems. The approach to the solution of these equations, and hence the construction of suitable methods, that we will describe in this paper, is based on computation of the coefficients of the stability polynomial by a variant of the Fourier series method and solving the resulting systems of polynomial equations by least squares minimization. Examples of explicit and implicit methods of order 5 and 6 are given which are appropriate for nonstiff or stiff differential systems in a sequential computing environment. The coefficients of these methods were obtained numerically with the aid of lmdif.f and lmder.f from MINPACK. These programs minimize the sum of the squares of nonlinear functions by a modification of the Levenberg-Marquardt algorithm. The derived explicit and implicit methods have the same stability properties as explicit Runge-Kutta and SDIRK methods, respectively, of the same order. (C) 1998 IMACS/Elsevier Science B.V.