| Abstract: |
| Many mathematical models in science and technology are based on first order symmetric hyperbolic systems of differential equations whose solutions must satisfy certain constraints (e.g., in electromagnetism, magnetohydrodynamics, and general relativity). When the models are restricted to bounded domains, the problem of well-posed, constraint-preserving boundary conditions arises naturally. However, for numerical solutions finding such boundary conditions may represent just a step in the right direction. Including the constraints as dynamical variables of a larger, unconstrained system associated to the original one could provide better numerical results, as the constraints are kept under control during evolution. One of the main goals of this talk is to present this idea in the case of constrained, constant-coefficient first order symmetric hyperbolic systems of differential equations subject to maximal nonnegative boundary conditions. As an example of application, a vector-valued wave equation with the constraint that the solution be divergence free is considered. Interestingly enough, on a smooth bounded domain, the set of constraint-preserving boundary conditions for this model problem involves the geometry of the boundary. If the domain is polyhedral, then this set of boundary conditions belongs to the well-known class of maximal nonnegative boundary conditions, and the general theoretical result applies. |
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