Optimal Constraint Projection for Hyperbolic Evolution Systems

TL;DR

This paper introduces optimal projection techniques for symmetric hyperbolic systems to effectively control constraint violations in numerical simulations, especially in black-hole backgrounds, by combining constraint projection with boundary conditions.

Contribution

It develops a method for optimal constraint projection in hyperbolic systems and demonstrates its effectiveness in numerical simulations involving scalar fields and black-hole backgrounds.

Findings

Constraint violations are effectively controlled by constraint projection.

Boundary violations are controlled by boundary-preserving conditions.

Solutions converge when combining projection with boundary conditions.

Abstract

Techniques are developed for projecting the solutions of symmetric hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field configuration to the ``nearest'' configuration in the constraint submanifold, where distances between configurations are measured with the natural metric on the space of dynamical fields. The construction and use of these projections is illustrated for a new representation of the scalar field equation that exhibits both bulk and boundary generated constraint violations. Numerical simulations on a black-hole background show that bulk constraint violations cannot be controlled by constraint-preserving boundary conditions alone, but are effectively controlled by constraint projection. Simulations also show that constraint violations entering through boundaries cannot be controlled by constraint projection alone, but are controlled by constraint-preserving boundary conditions. Numerical solutions to the pathological scalar field system are shown to converge to solutions of a standard representation of the scalar field equation when constraint projection and constraint-preserving boundary conditions are used together.