We consider possibly stiff and implicit systems of ordinary differential equations (ODEs). The major burden and computational load in the implementation of implicit Runge-Kutta (IRK) methods is generally the numerical solution of the arising nonlinear systems of equations. We show that the use of inexact simplified Newton methods for these nonlinear systems is very efficient. Linear systems of the simplified Newton method are solved approximately with a preconditioned linear iterative method. The preconditioner is based on the W-transformation of the RK coefficients and on the block-LU decomposition of the W-transformed simplified Jacobian. It is asymptotically exact for stiff systems. A new code based on those techniques, SPARK3, is shown to be effective on several problems.

MINISIMPOSIUM SESSION: 9. Solving linear and nonlinear systems in differential equations