Multi-block simulations in general relativity: high-order discretizations, numerical stability and applications

The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve a given problem under this grid structure, one must ensure the suitable transfer of information among the different grids involved. In this work, we discuss a technique that allows one to construct finite-difference schemes of arbitrary high order which are guaranteed to satisfy linear numerical and strict stability. The method relies on the use of difference operators satisfying summation by parts and penalty terms to transfer information between the grids. This allows the derivation of semi-discrete energy estimates for problems admitting such estimates at the continuum. We analyse several aspects of this technique when used in conjunction with high-order schemes and illustrate its use in one-, two- and three-dimensional numerical relativity model problems with non-trivial topologies, including truly spherical black hole excision