Summation-by-parts operators for non-simply connected domains

We construct fully discrete stable and accurate numerical schemes for solving partial differential equations posed on non-simply connected spatial domains. The schemes are constructed using summation-by-parts operators in combination with a weak imposition of initial and boundary conditions using the simultaneous approximation term technique. In the theoretical part, we consider the two dimensional constant coefficient advection equation posed on a rectangular spatial domain with a hole. We construct the new scheme and study well-posedness and stability. Once the theoretical development is done, the technique is extended to more complex non-simply connected geometries. Numerical experiments corroborate the theoretical results and show the applicability of the new approach and its advantages over the standard multi-block technique. Finally, an application using the linearized Euler equations for sound propagation is presented