Well-Posed Initial-Boundary Value Problem for a Constrained Evolution System and Radiation-Controlling Constraint-Preserving Boundary Conditions

TL;DR

This paper establishes a well-posed initial-boundary value problem for a constrained vector wave system, introducing new radiation-controlling boundary conditions that improve stability and eliminate ill-posed modes.

Contribution

It formulates a well-posed IBVP for the vector wave equation with divergence-free constraints and develops novel boundary conditions that enhance stability and constraint preservation.

Findings

The problem is proven well-posed with existence, uniqueness, and stability.

New boundary conditions satisfy the Kreiss condition and prevent polynomial growth of ill-posed modes.

Comparison shows the new conditions outperform standard boundary conditions in stability analysis.

Abstract

A well-posed initial-boundary value problem is formulated for the model problem of the vector wave equation subject to the divergence-free constraint. Existence, uniqueness and stability of the solution is proved by reduction to a system evolving the constraint quantity statically, i.e., the second time derivative of the constraint quantity is zero. A new set of radiation-controlling constraint-preserving boundary conditions is constructed for the free evolution problem. Comparison between the new conditions and the standard constraint-preserving boundary conditions is made using the Fourier-Laplace analysis and the power series decomposition in time. The new boundary conditions satisfy the Kreiss condition and are free from the ill-posed modes growing polynomially in time.