High-order narrow stencil finite-difference approximations of second-order derivatives involving variable coefficients

High-order-accuracy finite-difference approximations are developed for problems involving arbitrary variable coefficients in the second-order derivatives, e.g., the heat equation or turbulence modeling. The methods investigated are discretely conservative, use narrow stencils, and provide stable approximations for these problems. It is known that high-order finite-difference approximations for these types of equations using the chain rule approach may be inadequate for approximating partial differential equations with certain types of variable coefficients. The new approximations are constructed to alleviate this problem by requiring that the operators are stable when the variable coefficients are positive. Examples in heat-transfer problems with variable coefficients are shown to retain the designed order of accuracy and stability with lower error norms than the usual alternative discretizations. Finally, an application of the new stencils is presented for the large-eddy simulation of compressible turbulent flows.