Spectral Methods for Numerical Relativity

Abstract

Equations arising in general relativity are usually too complicated to be solved analytically and one must rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods in which, typically, the various functions are expanded in sets of orthogonal polynomials or functions. First, a theoretical introduction of spectral expansion is given with a particular emphasis on the fast convergence of the spectral approximation. We then present different approaches to solving partial differential equations, first limiting ourselves to the one-dimensional case, with one or more domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. We then present results obtained by various groups in the field of general relativity by means of spectral methods. Work, which does not involve explicit time-evolutions, is discussed, going from rapidly-rotating strange stars to the computation of black-hole-binary initial data. Finally, the evolution of various systems of astrophysical interest are presented, from supernovae core collapse to black-hole-binary mergers.

Figure 1

Function f = cos³ (πx/2) + (x + 1)³/8 (black curve) and its best approximation of degree 2 (red curve). The blue arrows denote the four points where the maximum error is attained.

Figure 2

Lagrange cardinal polynomials (red curve) and on an uniform grid with N = 8. The black circles denote the nodes of the grid.

Figure 3

Function f = cos³ (πx/2) + (x +1)³/8 (black curve) and its interpolant (red curve)on a uniform grid of five nodes. The blue circles show the position of the nodes.

Figure 4

Function (black curve) and its interpolant (red curve) on a uniform grid of five nodes (left panel) and 14 nodes (right panel). The blue circles show the position of the nodes.

Figure 5

Same as Figure 4 but using a grid based on the zeros of Chebyshev polynomials. The Runge phenomenon is no longer present.

Figure 6

Function f = cos³ (πx/2) + (x + 1)³/8 (black curve) and its projection on Chebyshev polynomials (red curve), for N = 4 (left panel) and N = 8 (right panel).

Figure 7

Function f = cos³ (πx/2) + (x + 1)³/8 (black curve) and its interpolant I_Nf on Chebyshev polynomials (red curve), for N = 4 (left panel) and N = 6 (right panel). The collocation points are denoted by the blue circles and correspond to Gauss-Lobatto quadrature.

Figure 8

First Legendre polynomials, from P₀ to P₅.

Figure 9

First Chebyshev polynomials, from T₀ to T₅.

Figure 10

Maximum difference between f = cos³ (πx/2) + (x + 1)³/8 and its interpolant I_Nf, as a function of N.

Figure 11

Step function (black curve) and its interpolant, for various values of N.

Figure 12

Exact solution (64) of Equation (62) (blue curve) and the numerical solution (red curve) computed by means of a tau method, for N = 4 (left panel) and N = 8 (right panel).

Figure 13

Exact solution (64) of Equation (62) (blue curve) and the numerical solution (red curve) computed by means of a collocation method, for N = 4 (left panel) and N = 8 (right panel).

Figure 14

Exact solution (64) of Equation (62) (blue curve) and the numerical solution (red curve) computed by means of the Galerkin method, for N = 4 (left panel) and N = 8 (right panel).

Figure 15

The difference between the exact solution (64) of Equation (62) and its interpolant (black curve) and between the exact and numerical solutions for i) the tau method (green curve and circle symbols) ii) the collocation method (blue curve and square symbols) iii) the Galerkin method (red curve and triangle symbols).

Figure 16

Difference between the exact and numerical solutions of the following test problem. , with S(x < 0) = 1 and S(x > 0) = 0. The boundary conditions are u(x = −1) = 0 and u(x = 1) = 0. The black curve and circles denote results from the multidomain tau method, the red curve and squares from the method based on the homogeneous solutions, the blue curve and diamonds from the variational one, and the green curve and triangles from the collocation method.

Figure 17

Regular deformation of the [−1, 1] × [−1, 1] square.

Figure 18

Two sets of spherical domains describing a neutron star or black hole binary system. Each set is surrounded by a compactified domain of the type (89), which is not displayed

Figure 19

Definition of spherical coordinates (r, θ, φ) of a point M and associated triad , with respect to the Cartesian ones.

Figure 20

Regions of absolute stability for the Adams-Bashforth integration schemes of order one to four.

Figure 21

Regions of absolute stability for the Runge-Kutta integration schemes of order two to five. Note that the size of the region increases with order.

Figure 22

Eigenvalues of the first derivative-tau operator (124) for Chebyshev polynomials. The largest (in modulus) eigenvalue is not displayed; this one is real, negative and goes as O(N²).

Figure 23

Behavior of the error in the solution of the differential equation (135), as a function of the parameter τ entering the numerical scheme (136).