
Abstract 
1 
Introduction 
2 
Notation and Conventions 
3 
The InitialValue
Problem 

3.1 
Linear, constant coefficient problems 

3.2 
Linear problems
with variable coefficients 

3.3 
Quasilinear equations 

3.4 
Abstract evolution
operators 
4 
InitialValue 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 InitialBoundary
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
timestability 

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 multidomain
binary blackhole evolutions 

11.5 
Multidomain studies of accretion
disks around black holes 

11.6 
Finitedifference multiblock orbiting
binary blackhole simulations 
12 
Acknowledgements 

References 

Footnotes 

Figures 

Tables 