Numerical Methods for the Einstein Equations in Null Quasi-Spherical Coordinates

• Published in: SIAM journal on scientific computing Vol. 22; no. 3; pp. 917 - 950
• Main Authors: Bartnik, Robert, Norton, Andrew H.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 2001
• Subjects: Accuracy; Algorithms; Cauchy problems; Classical general relativity; Euclidean space; Exact sciences and technology; Fundamental problems and general formalism; Gauges; General relativity and gravitation; Gravitational waves; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Other topics in general relativity and gravitation; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Physics; Radiation; Sciences and techniques of general use; Simulation; Spacetime; Invariant; Total energy; Fourier transformation; Noise; Black holes; Numerical method; Spline; Convergence; Convolution; Evolution; Reconstruction; Transport equation; Trautman Bondi mass loss formula; Hyperbolic system; Spherical harmonics; Characteristic; Coordinates; Mass loss; Boundary conditions; Invariance principles; Representation; Spectral limit; Equation system; Mass; Interpolation; Algorithms; Elliptic equations; Method of lines; Differential operator; Differentiation; Transport; Einstein equation; Preconditioning; Finite difference method
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/S1064827599356171

We describe algorithms used in our construction of a fourth-order in time evolution for the full Einstein equations and assess the accuracy of some representative solutions. The scheme employs several novel geometric and numerical techniques, including a geometrically invariant coordinate gauge, which leads to a characteristic-transport formulation of the underlying hyperbolic system, combined with a "method of lines" evolution; convolution splines for radial interpolation, regridding, differentiation, and noise suppression; representations using spin-weighted spherical harmonics; and a spectral preconditioner for solving a class of first-order elliptic systems on S 2. Initial data for the evolution is unconstrained, subject only to a mild size condition. For sample initial data of "intermediate" strength (19% of the total mass in gravitational energy), the code is accurate to 1 part in 105, until null time z=55M when the coordinate condition breaks down.