Four Lectures on Numerical Relativity
The first lecture is devoted to the causal structure of Einstein’s evo lution equations. They are written as a first-order system of balance laws, which is shown to be hyperbolic when the time coordinate is chosen in an invariant algebraic way (maximal slicing is recovered as a limiting case). The second lecture deals with first-order flux-conservative systems. The propagation of characteristic fields in an inhomogeneous background is also considered, with a view on relativity applications.
In the third lecture, explicit finite-difference numerical methods are reviewed, with an accent on flux-conservative second-order methods. Stability conditions are derived in each case. Finally, in the last lecture, total-variation-diminishing (TVD) methods are considered. The case of an inhomogeneous characteristic speed is illustrated with the evolution of a spherically symmetric (1D) black hole.
KeywordsBlack Hole Causal Structure Contact Discontinuity Advection Equation Characteristic Speed
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