约束条件和数值积分  

Constraints and Numerical Integration

作　　者：伍歆[1] 黄天衣[2] 

机构地区：[1]南昌大学理学院,南昌330047 [2]南京大学天文系,南京210093

出　　处：《天文学报》2004年第3期310-319,共10页Acta Astronomica Sinica

基　　金：国家自然科学重点基金(10233020);国家自然科学青年基金(10303001);南昌大学基础研究基金资助项目

摘　　要：自治的哈密顿系统存在约束条件,例如能量积分或广义相对论中的4速度大小为常数.它能否在数值积分过程中始终满足将直接影响数值稳定性.在牛顿力学中哈密顿系统的动能一般为椭圆型,直接运用约束条件对方程进行降阶存在开平方判断正负号的困难,导致应用高精度的经典数值积分器时能量存在耗散.然而相对论力学的度规为双曲型,利用约束条件有可能实行方程降阶.在时空具有一定对称性的情况下,能够找到整个时空的一个全局变换使变换后的度规的主对角线某一元素为零,于是从约束方程中不需开平方能够解出某一动量,顺利实现运动方程的降阶.相对论力学中另一个可以降阶的模型是Mixmaster宇宙模型.数值实验表明将经典算法用于降阶后的运动方程能够严格地满足约束,但不一定能保持辛结构.In an autonomous Hamiltonian system, tiere always exists at least one constraint, namely the energy integral or a constant magni ;ude of the 4-velocity in the case of general relativity. The constraint should bring better r umerical stability if it can be kept step by step in the process of numerical integration. In Newtonian mechanics, the order of equations of motion can not be decreased by use of the (onstraint in most cases, because its kinetic energy is usually in an elliptic pattern and one would meet difficulty in the operation of order decreasing. However, metrics in general relativity are in hyperbolic patterns. In particular, there exists a global transformation so that at least one element of the main diagonal vanishes when the spacetime bears some symmetries. As a reslut, a constraint can be solved for a certain velocity or a momentum without any difficulty, then the order of the equations of motion can be decreased. Similarly, this technique can also be applied to the evolution of Mixmaster universe. It is shown that this technique can raise precision and improve numerical stability dramatically even a classical integrator is adopted, although it might not keep the symplectic structure of the system.

关 键 词：约束条件 数值积分 哈密顿系统 广义相对论 Mixmaster宇宙模型 

分 类 号：P131[天文地球—天体力学]