Control Strategies for the Iterative Solution of Nonlinear Equations in ODE Solvers

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Abstract
In the numerical solution of ODEs by implicit time-stepping methods, a system of (nonlinear) equations has to be solved each step. It is common practice to use fixed-point iterations or, in the stiff case, some modified Newton iteration. The convergence rate of such methods depends on the stepsize. Similarly, a stepsize change may force a refactorization of the iteration matrix in the Newton solver. This paper develops new strategies for handling the iterative solution of nonlinear equations in ODE solvers. These include automatic switching between fixed-point and Newton iterations, investigating the "optimal" convergence rate with respect to total work per unit step, a strategy for when to reevaluate the Jacobian, a strategy for when to refactorize the iteration matrix, coordination with stepsize control. Examples will be given, showing that the new overall strategy works efficiently. In particular, the new strategy admits a restrained stepsize variation without refactorizations, thus permitting the use of a smoother stepsize sequence. The strategy is of equal importance for Runge--Kutta and multistep methods.