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