Advanced Boundary Conditions for Hyperbolic Systems

Numerical simulation of hyperbolic systems remains a challenge, particularly in the case of complex geometries. In particular, the need to construct meshes for complicated geometries is a bottleneck in many cases. This is especially evident when doing rapid prototyping and design optimization, where generating a new mesh for every trial geometry is prohibitive. These difficulties can be obviated by employing an embedded/immersed boundary method, in which boundary conditions are enforced weakly.

In my Ph.D. work, a new Nitsche-type approach is proposed for the weak enforcement of Dirichlet and Neumann boundary conditions in the context of time-domain wave propagation problems in mixed form. A peculiar feature of the proposed method is that, due to the hyperbolic structure of the problem considered, two penalty parameters are introduced, corresponding to Dirichlet and Neumann conditions, respectively. A stability and convergence estimate is also provided, in the case of a discontinuous-in-time Galerkin space–time integrator. The spatial discretization used is based on a stabilized method with equal order interpolation for all solution components. In principle, however, the proposed methodology is not confined to stabilized methods. An extensive set of tests are provided to validate the robustness and accuracy of the proposed approach.

The proposed Nitsche method is then extended to embedded domain computations of hyperbolic systems, using as models the equations of acoustic wave propagation and shallow water flows, through Shifted Boundary Method (SBM). The SBM belongs to the class of surrogate/approximate boundary algorithms and is based on the idea of shifting the location where boundary conditions are applied from the true to a surrogate boundary. Accordingly, boundary conditions, enforced weakly, are appropriately modified to preserve optimal error convergence rates. Accuracy, stability and robustness of the proposed method are tested by means of an extensive set of computational experiments for the acoustic wave propagation equations and shallow water equations. Comparisons with standard weak boundary conditions imposed on grids that conform to the geometry of the computational domain boundaries are also presented.