Initial Conditions for Numerical Relativity -- Introduction to numerical methods for solving elliptic PDEs

TL;DR

This paper introduces numerical methods for solving elliptic PDEs crucial in initial data setup and monitoring in numerical relativity, providing practical examples and code for beginners in the field.

Contribution

It offers an accessible overview and implementation guidance for elliptic PDE solutions in numerical relativity, emphasizing initial data preparation and black hole evolution.

Findings

Provides sample C++ and Fortran90 codes for elliptic PDEs

Demonstrates numerical solution techniques with simple examples

Highlights importance of elliptic equations in physics and relativity

Abstract

Numerical relativity became a powerful tool to investigate the dynamics of binary problems with black holes or neutron stars as well as the very structure of General Relativity. Although public numerical relativity codes are available to evolve such systems, a proper understanding of the methods involved is quite important. Here we focus on the numerical solution of elliptic partial differential equations. Such equations arise when preparing initial data for numerical relativity, but also for monitoring the evolution of black holes. Because such elliptic equations play an important role in many branches of physics, we give an overview of the topic, and show how to numerically solve them with simple examples and sample codes written in C++ and Fortran90 for beginners in numerical relativity or other fields requiring numerical expertise.