VODE: A Variable-Coefficient ODE Solver

Abstract

VODE is a new initial value ODE solver for stiff and nonstiff systems. It uses variable-coefficient Adams-Moulton and Backward Differentiation Formula (BDF) methods in Nordsieck form, as taken from the older solvers EPISODE and EPISODEB, treating the Jacobian as full or banded. Unlike the older codes, VODE has a highly flexible user interface that is nearly identical to that of the ODEPACK solver LSODE. In the process, several algorithmic improvements have been made in VODE, aside from the new user interface. First, a change in stepsize and/or order that is decided upon at the end of one successful step is not implemented until the start of the next step, so that interpolations performed between steps use the more correct data. Second, a new algorithm for setting the initial stepsize has been included, which iterates briefly to estimate the required second derivative vector. Efficiency is often greatly enhanced by an added algorithm for saving and reusing the Jacobian matrix J, as it occurs in the Newton matrix, under certain conditions. As an option, this Jacobian-saving feature can be suppressed if the required extra storage is prohibitive. Finally, the modified Newton iteration is relaxed by a scalar factor in the stiff case, as a partial correction for the fact that the scalar coefficient in the Newton matrix may be out of date. The fixed-leading-coefficient form of the BDF methods has been studied independently, and a version of VODE that incorporates it has been developed. This version does show better performance on some problems, but further tuning and testing are needed to make a final evaluation of it. Like its predecessors, VODE demonstrates that multistep methods with fully variable stepsizes and coefficients can outperform fixed-step-interpolatory methods on problems with widely different active time scales. In one comparison test, on a one-dimensional diurnal kinetics-transport problem with a banded internal Jacobian, the run time for VODE was 36 percent lower than that of LSODE without the J-saving algorithm and 49 percent lower with it. The fixed-leading-coefficient version ran slightly faster, by another 12 percent without J-saving and 5 percent with it. All of the runs achieved about the same accuracy.