Wilson , Grant J. This book presents an overview of the computational framework in which calculations of relativistic hydrodynamics have been developed. It summarizes the jargon and methods used in the field, and provides illustrative applications to real physical systems. The authors explain how to break down the complexities of Einstein's equations and fluid dynamics, stressing the viability of the Euler-Lagrange approach to astrophysical problems.

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In the years that followed, not only did computers become more powerful, but also various research groups developed alternate techniques to improve the efficiency of the calculations. With respect to black hole simulations specifically, two techniques were devised to avoid problems associated with the existence of physical singularities in the solutions to the equations: 1 Excision, and 2 the "puncture" method.

### Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity

In addition the Lazarus group developed techniques for using early results from a short-lived simulation solving the nonlinear ADM equations, in order to provide initial data for a more stable code based on linearized equations derived from perturbation theory. More generally, adaptive mesh refinement techniques, already used in computational fluid dynamics were introduced to the field of numerical relativity.

In the excision technique, which was first proposed in the late s, [12] a portion of a spacetime inside of the event horizon surrounding the singularity of a black hole is simply not evolved. In theory this should not affect the solution to the equations outside of the event horizon because of the principle of causality and properties of the event horizon i.

## Numerical Hydrodynamics in Special Relativity

Thus if one simply does not solve the equations inside the horizon one should still be able to obtain valid solutions outside. One "excises" the interior by imposing ingoing boundary conditions on a boundary surrounding the singularity but inside the horizon. While the implementation of excision has been very successful, the technique has two minor problems.

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The first is that one has to be careful about the coordinate conditions. While physical effects cannot propagate from inside to outside, coordinate effects could. For example, if the coordinate conditions were elliptical, coordinate changes inside could instantly propagate out through the horizon. This then means that one needs hyperbolic type coordinate conditions with characteristic velocities less than that of light for the propagation of coordinate effects e.

The second problem is that as the black holes move, one must continually adjust the location of the excision region to move with the black hole. The excision technique was developed over several years including the development of new gauge conditions that increased stability and work that demonstrated the ability of the excision regions to move through the computational grid.

In the puncture method the solution is factored into an analytical part, [20] which contains the singularity of the black hole, and a numerically constructed part, which is then singularity free. This is a generalization of the Brill-Lindquist [21] prescription for initial data of black holes at rest and can be generalized to the Bowen-York [22] prescription for spinning and moving black hole initial data. Until , all published usage of the puncture method required that the coordinate position of all punctures remain fixed during the course of the simulation.

## Numerical Relativistic Hydrodynamics: Finite-Difference Methods - Oxford Scholarship

Of course black holes in proximity to each other will tend to move under the force of gravity, so the fact that the coordinate position of the puncture remained fixed meant that the coordinate systems themselves became "stretched" or "twisted," and this typically lead to numerical instabilities at some stage of the simulation. In researchers demonstrated for the first time the ability to allow punctures to move through the coordinate system, thus eliminating some of the earlier problems with the method. This allowed accurate long-term evolutions of black holes.

The Lazarus project — was developed as a post-Grand Challenge technique to extract astrophysical results from short lived full numerical simulations of binary black holes.

It combined approximation techniques before post-Newtonian trajectories and after perturbations of single black holes with full numerical simulations attempting to solve General Relativity field equations. The Lazarus approach, in the meantime, gave the best insight into the binary black hole problem and produced numerous and relatively accurate results, such as the radiated energy and angular momentum emitted in the latest merging state, [26] [27] the linear momentum radiated by unequal mass holes, [28] and the final mass and spin of the remnant black hole.

Adaptive mesh refinement AMR as a numerical method has roots that go well beyond its first application in the field of numerical relativity.

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Mesh refinement first appears in the numerical relativity literature in the s, through the work of Choptuik in his studies of critical collapse of scalar fields. In the past few years, hundreds of research papers have been published leading to a wide spectrum of mathematical relativity, gravitational wave, and astrophysical results for the orbiting black hole problem. This technique extended to astrophysical binary systems involving neutron stars and black holes, [38] and multiple black holes. From Wikipedia, the free encyclopedia.

Der Feldgleichungen der Gravitation. In Witten, L.

Gravitation: An Introduction to Current Research. James R. Wilson , Grant J. This book presents an overview of the computational framework in which calculations of relativistic hydrodynamics have been developed. It summarizes the jargon and methods used in the field, and provides illustrative applications to real physical systems.