Overset grids provide an efficient and flexible framework for implementing high-order finite difference methods to simulate compressible viscous flows over complex geometries. Although overset methods have been widely used to solve time-dependent partial differential equations, very few proofs of stability exist for them. In practice, the interface treatments for overset grids are stabilized by adding numerical dissipation without any underlying theoretical analysis, impacting the accuracy and the conservation properties of the original method. In this work we discuss the construction of a provably time-stable and conservative method for solving hyperbolic problems on overset grids as well as their extension to solve the compressible Navier-Stokes equations. The proposed method uses interface treatments based on the simultaneous approximation term penalty method, and derivative approximations that satisfy the summation-by-parts property. Two cases of the method are analyzed. In the first case, no artificial dissipation is used and an eigenvalue analysis of the system matrix is performed to establish time-stability. The eigenvalue analysis approach for proving stability fails when the system matrix is not of a block triangular structure; therefore, we investigate the second case of the method where a localized numerical dissipation term is added to allow the use of energy method for stability proof. A framework for examining the conservation properties of the proposed method is discussed. Error analyses are performed to determine the order of interpolation that retains the accuracy of spatial finite difference operator.
The performance of the proposed method is assessed against the commonly used approach of injecting the interpolated data onto each grid. Several one-, two- and three-dimensional, linear and non-linear numerical examples are presented to confirm the stability and the accuracy of the methods. The extension of the method to solve the three-dimensional compressible Navier-Stokes equations on curvilinear grids is examined by performing a large-eddy simulation of flow over a cosine-shaped hill
