The use of integrals in numerical integrations of theN-body problem
The use of integrals in numerical integrations of theN-body problem
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DOI：
10.1007/bf00649193
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发表时间：
1971
影响因子：
1.9
通讯作者：
P. Nacozy
中科院分区：
文献类型：
--
作者：
P. Nacozy
具有积分的微分方程组的数值积分通常通过使用积分来减少自由度数或通过使用积分作为结果解的部分检查来实现，保留原始自由度数。这里介绍积分的另一种用途。如果尚未使用积分来简化系统，则可以通过在每个积分步骤对解应用校正的方法将数值积分的解限制为保留在积分表面上。通过在最小二乘法中使用积分的线性化形式来确定校正。给出了将该方法应用于 25 个物体的引力系统的数值积分的结果。结果表明，使用该方法精确满足能量、角动量和质心的积分，比不精确满足积分时获得的解更准确，且计算时间更少。相对精度是通过校正和未校正解的前向和后向积分以及通过与使用减小的步长的更准确的积分进行比较来确定的。
The numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes.