VODE - A VARIABLE-COEFFICIENT ODE SOLVER
VODE - A VARIABLE-COEFFICIENT ODE SOLVER
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DOI：
10.1137/0910062
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发表时间：
1989-09-01
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通讯作者：
HINDMARSH, AC
中科院分区：
文献类型：
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作者：
BROWN, PN;BYRNE, GD;HINDMARSH, AC
VODE是一种适用于刚性和非刚性系统的新的初值微分方程组求解器。它使用Nordsieck形式的变系数Adams-Moulton和反向微分公式(BDF)方法，该方法取自较早的求解器SEIPSE和EPISODEB，将雅可比矩阵视为完整的或带状的。与旧的代码不同，Vode有一个高度灵活的用户界面，几乎与ODEPACK求解器LSODE的用户界面相同。在这个过程中，除了新的用户界面外，Vode还在算法上进行了一些改进。首先，在一个成功的步骤结束时决定的步长和/或顺序的改变直到下一步的开始才被实施，从而在步骤之间执行的内插使用更正确的数据。其次，提出了一种新的设置初始步长的算法，该算法简单地迭代以估计所需的二阶导数向量。在某些条件下，当雅可比矩阵出现在牛顿矩阵中时，通过增加保存和重复使用雅可比矩阵J的算法，效率通常会大大提高。作为一种选择，如果所需的额外存储是令人望而却步的，则可以抑制这种雅可比节省特征。最后，针对牛顿矩阵中标量系数可能过时的问题，对刚性情况下的修正牛顿迭代法进行了标量因子的松弛，并对固定前导系数形式的BDF方法进行了独立的研究，开发了包含固定前导系数形式的VODE。这个版本确实在一些问题上表现出了更好的性能，但还需要进一步的调整和测试才能对其进行最终评估。与以前的版本一样，Vode证明了完全可变步长和系数的多步法在有效时间尺度差异很大的问题上的性能优于固定步长插值法。在一次对比测试中，在一个带有带状内部雅可比的一维全天动力学-传输问题上，VODE的运行时间比没有J-Saving算法的LSODE减少了36%，使用J-Saving算法时减少了49%。固定领先系数版本的运行速度略快，在没有J储蓄的情况下又快了12%，在J储蓄的情况下又快了5%。所有的运行都达到了大致相同的准确度。
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 matrixJ, 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factor in the stiff case, as a partial correction for the fact that the scalar coefficient in the Newton matrix may be out of date.The fixed-leading-coefficient form of the BDF methods has been studied independently, and a version of VODE that incorporates it has been developed.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.