Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

When discretizing equations with second-order derivatives, like the Navier-Stokes equations, the boundary and interior penalties play a critical role in terms of the conservation, consistency, energy stability, and adjoint consistency of summation-by-parts (SBP) methods. While interior penalties for tensor-product SBP operators have been studied, they have not been investigated in the context of multidimensional SBP operators. This paper presents a general discretization for analyzing interior penalties for multidimensional SBP operators that can be used to obtain several favorable properties. Under this discretization, we construct a stable and adjoint consistent high-order finite difference scheme for linear elliptic equations with a constant diffusion coefficient. Specifically, the equations are first discretized using multidimensional SBP operators with interior and boundary penalty matrices to be determined. Then, taking advantage of the properties of SBP operators, the analyses are generalized from those in the finite element literature and become entirely algebraic in nature. That is, the analyses give rise to explicit conditions on the penalty matrix coefficients and these conditions are free of integral. To validate these conditions numerically, several test cases are conducted.