Iterative Solution of Nonlinear Equations for Spark Methods Applied to Daes

We consider a broad class of systems of implicit differential-algebraic equations (DAEs) including the equations of mechanical systems with holonomic and nonholonomic constraints. We approximate numerically the solution to these DAEs by applying a class of super partitioned additive Runge-Kutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the combination of Lobatto IIIA-B-C-C∗-D methods, are crucial to deal properly with the presence of constraints and algebraic variables. A major difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. To solve these systems we make use of inexact modified Newton iterations. Linear systems of the modified Newton method are solved approximately with a preconditioned linear iterative method. The preconditioner is based in part on the W-transformation of the SPARK coefficients and also on some specific linear transformations to the original system of nonlinear equations. These linear transformations rely heavily on certain properties of the SPARK coefficients. For nonstiff DAEs the preconditioner requires only one matrix decomposition of the same dimension as the DAEs.