A stopping criterion for the Newton-Raphson method in implicit multistep integration algorithms for nonlinear systems of ordinary differential equations

Abstract

In the numerical solution of ordinary differential equations, certain implicit linear multistep formulas, i.e. formulas of type ∑ ^k _{j =0} α _j x _{n + j} - h ∑ ^k _{j =0} β _j x _{n + j} = 0, (1) with β _k > ≠ 0, have long been favored because they exhibit strong (fixed- h ) stability. Lately, it has been observed [1-3] that some special methods of this type are unconditionally fixed- h stable with respect to the step size. This property is of great importance for the efficient solution of stiff [4] systems of differential equations, i.e. systems with widely separated time constants. Such special methods make it possible to integrate stiff systems using a step size which is large relative to the rate of change of the fast-varying components of the solution.