Time step adaptivity in the method of Dahlquist, Liniger and Nevanlinna

Abstract Dahlquist, Liniger and Nevanlinna devised a family of one-leg two-step methods (DLN) that is second order, A- and G- stable for arbitrary, non-uniform time steps. The DLN method thus has strong potential for use in adaptive codes, but its adaptive step size selection is little explored. This report develops two approaches for the efficient local error estimation in the DLN method, and tests their use in a standard adaptivity framework. Many methods of error estimation are possible; herein we focus on two complementary estimators which involve minimal extra storage and computations. First we evaluate the local truncation error of the DLN method by Milne’s device, using the difference between the solution of the DLN method and the solution of a variable-step, explicit, second-order Adams-Bashforth-like method. Second, we use a recent refactorization of the DLN method, which eases implementation of DLN in legacy codes, to obtain an effective error estimation at no extra cost. We perform a number of numerical tests, comparing the two time adaptive DLN algorithms with some standard numerical ODE packages. Our tests indicate that the adaptive DLN method, with error estimated by Milne’s device, is an efficient and reliable method, even for stiff and unstable problems.