Conservative time integrators of arbitrary order for skew-symmetric finite-difference discretizations of compressible flow

•We examine conservative time integrators for the skew-symmetric flow equations.•A formulation of the Navier–Stokes equations in · variables is introduced.•Gauss collocation schemes lead to a norm-conserving discretization.•A fully conservative scheme of arbitrary order in space and time is presented.•Numerical test cases of isotropic turbulence are evaluated.

Skew-symmetric discretizations of the Navier–Stokes equations avoid the introduction of artificial numerical damping by first principles and are thus attractive for the simulation of turbulence and flow induced sound. For direct numerical simulations a high discretization order in space and time is essential to obtain high quality results at affordable cost. While skew-symmetric and fully conservative schemes of high spatial discretization order are available, the time integrators respecting skew-symmetry and conservation are of second order only. We present a class of time integrators for skew-symmetrical schemes for compressible flow, which allow an arbitrary order of accuracy. We also show that these schemes are discretely norm-conserving. Test cases for isotropic turbulence are performed. A comparison with standard spatial and temporal discretization completes our survey.