Some Mathematical And Numerical Questions Connected With First And Second Order Time Dependent Systems Of Partial Differential Equations

摘要

There is a tendency to write the equations of general relativity as a firstorder symmetric system of time dependent partial differential equations.However, for numerical reasons, it might be advantageous to use a second orderformulation like one obtained from the ADM equations. Unfortunately, the typeof the ADM equations is not well understood and therefore we shall discuss, inthe next section, the concept of wellposedness. We have to distinguish betweenweakly and strongly hyperbolic systems. Strongly hyperbolic systems are wellbehaved even if we add lower order terms. In contrast; for every weaklyhyperbolic system we can find lower order terms which make the problem totallyillposed. Thus, for weakly hyperbolic systems, there is only a restricted classof lower order perturbations which do not destroy the wellposedness. Toidentify that class can be very difficult, especially for nonlinearperturbations. In Section 3 we will show that the ADM equations, linearizedaround flat with constant lapse function and shift vector, are only weaklyhyperbolic. However, we can use the trace of the metric as a lapse function tomake the equations into a strongly second order hyperbolic system. Using simplemodels we shall, in section 4, demonstrate that approximations of second orderequations have better accuracy properties than the corresponding approximationsof first order equations. Also, we avoid spurious waves which travel againstthe characteristic direction. In the last section we discuss some difficultiesconnected with the preservation of constraints.