Living Reviews in Relativity

"Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations"
Olivier Sarbach and Manuel Tiglio 

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1 Introduction
2 Notation and Conventions
3 The Initial-Value Problem
3.1 Linear, constant coefficient problems
3.2 Linear problems with variable coefficients
3.3 Quasilinear equations
3.4 Abstract evolution operators
4 Initial-Value Formulations for Einstein’s Equations
4.1 The harmonic formulation
4.2 The ADM formulation
4.3 The BSSN formulation
4.4 Other hyperbolic formulations
5 Boundary Conditions: The Initial-Boundary Value Problem
5.1 The Laplace method
5.2 Maximal dissipative boundary conditions
5.3 Absorbing boundary conditions
6 Boundary Conditions for Einstein’s Equations
6.1 The harmonic formulation
6.2 Boundary conditions for BSSN
6.3 Geometric existence and uniqueness
6.4 Alternative approaches
7 Numerical Stability
7.1 Definitions and examples
7.2 The von Neumann condition
7.3 The method of lines
7.4 Strict or time-stability
7.5 Runge–Kutta methods
7.6 Remarks
8 Spatial Approximations: Finite Differences
8.1 Polynomial interpolation
8.2 Finite differences through interpolation
8.3 Summation by parts
8.4 Stability
8.5 Numerical dissipation
8.6 Going further
9 Spatial Approximations: Spectral Methods
9.1 Spectral convergence
9.2 Some properties of orthogonal polynomials
9.3 Legendre and Chebyshev polynomials
9.4 Gauss quadratures and summation by parts
9.5 Discrete expansions and interpolation
9.6 Spectral collocation differentiation
9.7 The collocation approach
9.8 Going further, applications in numerical relativity
10 Numerical Boundary Conditions
10.1 Outer boundary conditions
10.2 Interface boundary conditions
10.3 Going further, applications in numerical relativity
11 Domain Decomposition
11.1 The power and need of adaptivity
11.2 Adaptive mesh refinement for BBH in higher dimensional gravity
11.3 Adaptive mesh refinement and curvilinear grids
11.4 Spectral multi-domain binary black-hole evolutions
11.5 Multi-domain studies of accretion disks around black holes
11.6 Finite-difference multi-block orbiting binary black-hole simulations
12 Acknowledgements
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