A conservative, skew-symmetric finite difference scheme for the compressible Navier–Stokes equations

• A conservative and skew-symmetric scheme for compressible flows is shown.
• It is a very simple to program finite difference scheme compared with standard SSS.
• It builds purely on standard derivatives and point-wise multiplication.
• Arbitrary order and wave number optimized central derivatives can be used.
• The scheme can handle shocks and acoustics.

We present a fully conservative, skew-symmetric finite difference scheme on transformed grids. The skew-symmetry preserves the kinetic energy by first principles, simultaneously avoiding a central instability mechanism and numerical damping. In contrast to other skew-symmetric schemes no special averaging procedures are needed. Instead, the scheme builds purely on point-wise operations and derivatives. Any explicit and central derivative can be used, permitting high order and great freedom to optimize the scheme otherwise. This also allows the simple adaption of existing finite difference schemes to improve their stability and damping properties. Numerical examples covering acoustic phenomena and shocks are presented.