Constrained Evolution of Black Hole Spacetimes

Matthew Anderson, University of Texas

The theory of relativity is a geometric theory of gravitation: gravitational forces are explained as the local curvature of spacetime. The Einstein equations describing spacetime curvature are expressed in a coordinate independent way. However, to solve the equations numerically we must use a coordinate-dependent form. When the Einstein equations are recast into a “3+1” form, separating space from time, two classes of equations result: evolution equations and constraint equations. The evolution equations are hyperbolic while the constraint equations are elliptic. Successfully solving the evolution equations should automatically ensure the satisfaction of the constraint equations, but this is rarely achieved. Using the CVODE integrator from the SUNDIALS suite and the elliptic solving capabilities of PETSc, we solve both the evolution and constraint equations simultaneously to obtain long-term stability in black hole simulations. We present results from numerical simulations of constrained evolution for single and multiple black hole configurations.

Abstract Author(s): Matthew Anderson, Richard Matzner