VODE: A Variable-Coefficient ODE Solver

作者：Peter N. Brown, George D. Byrne, A.C. Hindmarsh · 发表于：SIAM Journal on Scientific and Statistical Computing · 年份：1989 · DOI：10.1137/0910062 · 被引用次数：1647 · 研究领域：Numerical methods for differential equations、Matrix Theory and Algorithms、Differential Equations and Numerical Methods

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factor in the stiff case, as a partial correction for the fact that the scalar coefficient in the Newton matrix may be out of date.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 develop...