Constraints and numerical integration

Details Constraints and numerical integration Constraints and numerical integration

Aug 2004

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2004acasn..45..310w&link_type=abstract

Acta Astronomica Sinica, vol. 45, no. 3, p. 310-319

Astronomy and Astrophysics

Astronomy

2

Celestial Mechanics: Numerical Integration, General Relativity, Mixmaster Universe

Scientific paper

In an autonomous Hamiltonian system, there always exists a constraint, namely the energy integral or a constant magnitude of the 4-velocity in relativistic dynamics. The constraint should bring better numerical stability if it can be kept step by step in the course of numerical integration. In Newtonian mechanics, the order of the equations of motion can not be reduced by use of the constraint in most cases, because its kinetic energy is usually an elliptic type of quadratic form and one would meet difficulty in the operation of order reducing. However, the metric in general relativity is hyperbolic. In particular, when the spacetime bears some symmetries there exists a global transformation so that at least one element in its main diagonal vanishes. As a result, the constraint can be solved for a certain velocity or a momentum without any difficulty, and then the order of the equations of motion can be reduced. 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.

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