Excision and avoiding the use of boundary conditions in numerical relativity

Abstract

A procedure for evolving hyperbolic systems of equations on compact computational domains with no boundary conditions was recently described in [1]. In that proposal, the computational grid is expanded in spacelike directions with respect to the outermost characteristic and initial data is imposed on the expanded grid boundary. We discuss a related method that removes the need for imposing boundary conditions: the computational domain is excised along a direction spacelike or tangent to the innermost going characteristic. We compare the two methods, and provide example evolutions from a code that implements the excision method: evolution of a massless self-gravitating scalar field in spherical symmetry. * jripley@princeton.edu 1 arXiv:1908.04234v2 [gr-qc]

Type II critical collapse on a single fixed grid: a gauge-driven ingoing boundary method

“…We presented a numerical method for gravitational collapse based on Cauchy evolution with an ingoing null boundary. The method is similar in spirit to the excision method of Ripley [14] but differs in that no grid points are removed from the computational domain; rather, the grid remains fixed and only the coordinates are adapted along with the evolution. This is achieved by adding a linear term to the shift vector that causes the coordinates to "zoom in" isotropically towards the centre.…”

“…Finally it should be stressed that this ingoing boundary method or the related method of Ripley [14] are not limited to studying critical collapse. One can also start with a standard Cauchy evolution with timelike boundary (where of course boundary conditions must be imposed) and switch to the ingoing boundary method at a certain time.…”

“…(12) and (13) for the scalar field do not require any boundary conditions. We specify Dirichlet boundary conditions on ψ for the Hamiltonian constraint (9) by evolving (14) at the outer boundary. The momentum constraint (8) does not require a boundary condition as this is already fixed by demanding the solution to be regular at the origin.…”

“…At the advanced time the radial ordinary differential equations (ODEs) (8)- (11) are solved in this order forK r r , ψ, α and β (notice they form a hierarchy). Dirichlet boundary values for ψ and α are supplied by evolving (14) and (17) at the outer boundary along with the other evolution equations, and the boundary condition for β is (16).…”

“…Related to Bieri et al's scheme is the "excision method" proposed by Ripley [14], whereby the computational domain is excised along a surface that is spacelike or tangent to an ingoing characteristic of the boundary on the initial slice. Again, no boundary conditions need to be impose at the outer boundary because in this case all characteristics leave the computational domain.…”