Inexact Simplified Newton Iterations For Implicit Runge-Kutta Methods

We consider possibly sti# and implicit systems of ordinary di#erential equations (ODEs). The major di#culty and computational bottleneck in the implementation of fully implicit Runge--Kutta (IRK) methods resides in the numerical solution of the resulting systems of nonlinear equations. To solve those systems we show that the use of inexact simplified Newton methods is e#cient. Linear systems of the simplified Newton method are solved approximately with a preconditioned linear iterative method. Su#cient conditions ensuring local convergence of the inexact simplified Newton method for general nonlinear equations are given. The preconditioner that we use is based on the W-transformation of the RK coe#cients and on the block-LU decomposition of the simplified Jplified after W-transformation. A new code based on those techniques, SPARK3, is shown to be e#ective on two problems; the first one is a linear convection-di#usion problem and the second one a reaction-di#usion problem