Controlling the Growth of Constraints in Hyperbolic Evolution Systems

TL;DR

This paper explores two methods to control exponential growth of constraint violations in hyperbolic evolution systems, demonstrating their effectiveness in numerical Maxwell equations, with boundary conditions proving more successful than active control.

Contribution

It introduces and compares two novel methods for constraint growth control in hyperbolic systems, emphasizing the effectiveness of boundary conditions over active adjustments.

Findings

Constraint preserving boundary conditions are more effective than active constraint control.

The methods successfully reduce constraint growth in 3D Maxwell simulations.

Boundary conditions significantly improve numerical stability.

Abstract

Motivated by the need to control the exponential growth of constraint violations in numerical solutions of the Einstein evolution equations, two methods are studied here for controlling this growth in general hyperbolic evolution systems. The first method adjusts the evolution equations dynamically, by adding multiples of the constraints, in a way designed to minimize this growth. The second method imposes special constraint preserving boundary conditions on the incoming components of the dynamical fields. The efficacy of these methods is tested by using them to control the growth of constraints in fully dynamical 3D numerical solutions of a particular representation of the Maxwell equations that is subject to constraint violations. The constraint preserving boundary conditions are found to be much more effective than active constraint control in the case of this Maxwell system.