High-Order Compact-Stencil Summation-By-Parts Operators for the Compressible Navier-Stokes Equations

A general framework is presented for deriving minimum-stencil high-order summationby-parts finite-difference operators for the second derivative with variable coefficients, for orders of accuracy 3 through 6. These operators can be used to construct time-stable numerical schemes with simultaneous approximation terms to weakly impose boundary conditions. The derivation of these operators leads to various free parameters which can be used for optimization of the operator about criteria such as spectral radius and truncation error. The operators are 2p accurate on the interior, where the interior stencil has 2p+1 nodes, but p accurate at the boundaries. Nonetheless, for purely parabolic problems, they can be shown to be p+2 globally accurate. However, for the Navier-Stokes equations, the continuity equation renders the method p + 1 accurate. We present a novel means of circumventing this degradation in accuracy by using p + 2 globally accurate operators for the continuity equation, and we prove that the new discretization remains amenable to the energy method, a necessary condition to prove time stability. Numerical tests on the one-dimensional linear convection-diffusion equation and the oneand three-dimensional Navier-Stokes equations using the method of manufactured solutions are used for verification and characterization studies. We show that for the Navier-Stokes equations using a p+ 2 globally accurate first derivative for the continuity equation substantially increases the accuracy benefits of the minimum-stencil operator.