## Computational methods for Einstein's equations

### Participants

- Ken Mattsson
- Manuel Tiglio, University of Maryland
- Oscar Reula, University of Cordoba
- Florencia Parisi, University of Cordoba

### Research

This project is focused on the numerical solutions of Einstein's equations, which describe processes such as binary black hole collisions, supernovas, pulsars, and the big bang. The outcome of this kind of simulation is considered to be crucial for the successful detection and interpretation of gravitational waves, expected to be measured by laser interferometers such as LIGO, GEO600, LISA, and others. In turn, measurement of gravitational waves will constitute a strong, direct verification of Einstein's theory, and open a new window to the universe. The construction of these numerical solutions requires large-scale computations and research on a variety of physical, mathematical, numerical, and scientific computing issues. The simulation of black holes has proved to be a very difficult computational problem.

The computational domain is often enormous compared with the wavelengths, which means that the generated gravity-waves (the solution to Einstein's equations) have to travel very long distances (or correspondingly very long times). As a result, highly stable and accurate numerical methods that can be efficiently implemented in non-trivial geometries and on parallel super-computers are required. High-order finite difference methods fulfill these requirements, and will be the method used in this project. The issue of constructing highly stable and accurate numerical algorithms, including the boundaries, of the Einstein's equation written on their natural second order form has not yet been properly addressed and is the main goal of this project.

The energy method is a common technique to derive well-posedness for initial-boundary value problems (IBVP). A very powerful way of obtaining provable stable numerical approximations is to exactly mimic the underlying continuous energy estimate of the IBVP. To accomplish this, finite difference operators that satisfy a Summation-By-Parts (SBP) formula have to be utilized. For the Einstein's equations written on second order form, the regular energy estimate fails in the most interesting applications, which have required the introduction of a modified energy estimate. The existing SBP operators are based on the regular energy estimates, which means that they are not suitable for deriving stable algorithms of Einstein's equations.

The specific goals of the project are the following:

- The construction of new very high-order accurate SBP operators based on the modified energy estimate.
- The development of (based on the new SBP operators) a highly stable and accurate solver for the simulation of Einstein's equations on second order form, implemented on a parallel supercomputer.
- The development of post-processing techniques, in order to interpret and validate the measurements from ground- and space-based gravitational wave detectors such as LIGO, GEO600, LISA, and others.

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### Publications

- Ken Mattsson,
*Stable second-order formulation of Einstein's equations*. Center for Turbulence Research Annual Research Briefs, Stanford University, 2007.

- Ken Mattsson and Florencia Parisi,
*Stable and accurate second-order formulation of the shifted wave equation*. In Communications in Computational Physics, volume 7, pp 103-137, 2010.

**Contact**: , Ken Mattsson