Towards provable energy-stable overset grid methods using sub-cell summation-by-parts operators

Abstract

Overset grid methods handle complex geometries by overlapping simpler, geometry-fitted grids to cover the original, more complex domain. However, ensuring their stability---particularly at high orders---remains a theoretical challenge: although overset grid methods perform robustly in extensive practical use, general stability proofs are not available. In this work, we address this gap by developing a discrete counterpart to the recent well-posedness analysis of Kopriva, Gassner, and Nordstr\"om for continuous overset domain initial-boundary-value problems. To this end, we introduce the novel concept of sub-cell summation-by-parts (SBP) operators. These discrete derivative operators mimic integration by parts at a sub-cell level. By exploiting this sub-cell SBP property, we develop provably conservative and energy-stable overset grid methods for fixed one-dimensional overset domains that do not change with time or under grid refinement, providing a step toward stability proofs for overset grid methods based on the energy method.