Inexact Simplified Newton Iterations for Implicit Runge-Kutta Methods

We consider possibly stiff and implicit systems of ordinary differential equations (ODEs). The major difficulty 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 efficient. Linear systems of the simplified Newton method are solved approximately with a preconditioned linear iterative method. Sufficient 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 coefficients and on the block-LU decomposition of the simplified Jacobian after W-transformation. A new code based on those techniques, SPARK3, is shown to be effective on two problems; the first one is a linear convection-diffusion problem and the second one a reaction-diffusion problem.