Energy stable overset grid methods for hyperbolic problems

We investigate an alternative approach of proving energy stability for hyperbolic problems on overset (or overlapping) grids. Instead of bounding the time derivative of the discrete energy in a given norm, we look at the system matrix and show stability in terms of its eigenvalues. This approach is successful for hyperbolic problems since they have a characteristic direction of propagation which yields system matrices whose eigenvalues could be estimated. The schemes are based on summation-by-parts (SBP) operators and the simultaneous approximation term (SAT) procedure for boundary conditions and interface treatments. The SAT implementation requires the knowledge of the norm matrix in which the energy estimate of the method exists. Since the complexity of determining the norm matrix prompted this investigation, we use the norm in which the single domain scheme is time stable to construct SAT treatments for overset grid and then we check its stability. The proposed treatment is energy stable for scalar hyperbolic equations irrespective of the kind and location of interpolation. The scheme is also energy stable for systems of decoupled hyperbolic equations but not for a periodic system. We show that a numerical scheme from our previous work is time stable for the periodic system.