A NEW NUMERICAL ALGORITHM FOR EFFICIENTLY IMPLEMENTING IMPLICIT RUNGE-KUTTA METHODS

To efficiently implement implicit Runge-Kutta (IRK) methods for solving large scale stiff ordinary differential equation systems, this paper proposes a new numerical algorithm, which contains a new modified Newton iterative method for solving the nonlinear stage equations of IRK, and two new rules for controlling the updating of Jacobian matrices. A convergence analysis shows that the new modified Newton method has a faster rate of convergence than the simplified Newton method, which is widely used in implementing IRK, and the two new rules can significantly reduce the total number of Jacobian matrix evaluations while retaining a fast rate of convergence. The new algorithm is numerically studied in the implementation of a widely-used IRK scheme, the s-stage Radau IIA method with s = 3, 5, or 7, showing that the new modified Newton method can significantly increase the step size range and reduce the computer CPU time in solving a stiff equation compared to the current simplified Newton method. It is also adapted to the well-known Radau IIA program package RADAU to demonstrate the potential of the new algorithm in enhancing the performance of RADAU.

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