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| Important mathematical models involve hyperbolic systems of differential equations supplemented by constraint equations on infinite domains. In general, for the pure Cauchy problem one can prove that the constraints are preserved by the evolution. That is, the solution satisfies the constraints for all time whenever the initial data does (e.g., Maxwell's equations and Einstein's field equations in various hyperbolic formulations). Frequently, the numerical solutions to such evolution problems are computed on artificial space cutoffs because of the necessary boundedness of computational domains. Therefore, well-posed boundary conditions are needed at the artificial boundaries. Moreover, these boundary conditions have to be chosen in such a way that the numerical solution of the cutoff system approximates as best as possible the solution of the original problem on infinite domain, and this includes the preservation of constraints. In this talk, I will present a few ideas and techniques for finding constraint preserving boundary conditions for a large class of constrained hyperbolic systems. Then, I will talk about applications of the theoretical framework to certain models. |
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