Explicit Runge–Kutta Methods with Estimates of the Local Truncation Error
Explicit Runge–Kutta Methods with Estimates of the Local Truncation Error
复制标题
DOI：
10.1137/0715051
复制
发表时间：
1978-08
影响因子：
2.9
通讯作者：
J. Verner
中科院分区：
文献类型：
--
作者：
J. Verner
常微分方程组近似解的有效算法依赖于通过调整步长(可能还包括阶数)来控制误差估计。对于显式Runge-Kutta方法，通常使用局部截断误差的估计，目前最有效的估计可以是某些连续阶方法对的差。由于推导中间精度阶数的方法很困难，而且这种方法对许多具有中等公差要求的问题都是有效的，因此Fehlberg导出的方法变得特别著名。不幸的是，对于简化为求积求值的问题，Fehlberg方法给出的误差估计相同为零，因此对于至少部分属于这类问题的估计是不可靠的。这里开发的方法就是为了克服这个困难而设计的。该方法给出了任意高阶精度曲线的新方法。
Efficient algorithms for the approximate solution of ordinary differential equations rely on controlling estimates of the error through adjustment of stepsize (and possibly, of order). For explicit Runge-Kutta methods estimates of the local truncation error are normally used, and currently the most efficient estimates may be obtained as differences of certain pairs of methods of successive orders. Because of the difficulty of deriving methods of intermediate orders of accuracy, and because such methods are efficient for many problems with moderate tolerance requirements, the methods derived by Fehlberg have become particularly well-known. Unfortunately, for problems that reduce to the evaluation of quadratures, Fehlberg’s methods give error estimates which are identically zero, and hence the estimates are unreliable for problems that are at least partially of this type. The methods developed here are designed to overcome this difficulty. The approach yields new methods of arbitrarily high orders of accura...