Development of Generalized Summation-by-Parts Operators for the Second Derivative with Variable Coefficients

The generalized summation-by-parts (GSBP) framework enables the derivation of novel and potentially efficient provably stable high-order finite-difference operators, applicable on general nodal distributions. This paper explores the application of GSBP operators to the solution of partial differential equations with firstand second-derivative terms. Here, we investigate compatible and order-matched GSBP operators for the approximation of the second derivative with a variable coefficient. These operators are one order more accurate than the application of the first-derivative operator twice and more dissipative of underresolved modes. Several example operators are described in detail. To characterize the various operators, the steady linear convection-diffusion equation with a variable coefficient is solved.