Living Reviews in Relativity

"Spectral Methods for Numerical Relativity"
by
Philippe Grandclément and Jérôme Novak 

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Abstract
1 Introduction
1.1 About spectral methods
1.2 Spectral methods in physics
1.3 A simple example
2 Concepts in One Dimension
2.1 Best polynomial approximation
2.2 Interpolation on a grid
2.3 Polynomial interpolation
2.4 Usual polynomials
2.5 Spectral methods for ODEs
2.6 Multidomain techniques for ODEs
3 Multidimensional Cases
3.1 Spatial coordinate systems
3.2 Spherical coordinates and harmonics
3.3 Going further
4 Time-Dependent Problems
4.1 Time discretization
4.2 Imposition of boundary conditions
4.3 Discretization in space: stability and convergence
4.4 Fully-discrete analysis
4.5 Going further: High-order time schemes
5 Stationary Computations and Initial Data
5.1 Introduction
5.2 Single compact stars
5.3 Single black holes
5.4 Rings around black holes
5.5 Compact star binaries
5.6 Black-hole–binary systems
5.7 Black-hole–neutron-star binaries
5.8 Spacetimes with waves
5.9 Hyperboloidal initial data
6 Dynamic Evolution of Relativistic Systems
6.1 Single Stars
6.2 Vacuum and black hole evolutions
6.3 Binary systems
7 Conclusions
7.1 Strengths and weaknesses
7.2 Combination with other methods
7.3 Future developments
Open References References
Footnotes
Updates
Figures