Stable and Accurate Second-Order Formulation of the Shifted Wave Equation.

KEN MATTSSON; FLORENCIA PARISI

México DF

Conferencia; 19th International Conference on General Relativity and Gravitation; 2010

The present study is focused towards the numerical solution of Einstein’s equations, which describe processes such as binary black holes and neutron star collisions, and can be regarded as the first step towards a strictly stable simulation of the second-order formulation of Einstein’s equations in three spatial dimensions. Although these equations can be reduced to first-order symmetric hyperbolic form, this has the disadvantage of introducing auxiliary variables with their constraintsand boundary conditions, and is also less attractive from a computational point of view considering the efficiency and accuracy.For wave-propagation problems, the computational domain is often large compared to the wavelengths, which means that waves have to travel long distances over long times. As a result, high-order accurate time marching methods, as well as high-order spatially accurate schemes (atleast third-order) are required.In this work, high order finite difference approximations are derived for a onedimensionalmodel of the shifted wave equation written in second-order form, which is a model that capturesmost of the stability issues of the full 3-dimensional problem without introducing unnecessary complications . The domain is discretized using fully compatible summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a strictly stable and efficient scheme the faithfully reproduces the expected behavior of the wave. The analysis is verified by numerical simulations in one-dimension.