A high-order CFD method using successive differentiation

There has been a growing interest in higher-order spatial discretization methods as they can potentially give higher resolution accuracy at lower computational cost for the Direct Numerical Simulation (DNS) of vortex-dominated flows. Many of the modern high-order schemes use more degrees-of-freedom (DOF) in each cell to achieve high-order accuracy. This paper formulates and demonstrates a high-order (up to 4th order) correction method by using successive differentiation method. Unlike the popular Discontinuous Galerkin method, the present approach does not increase the degrees-of-freedom in each cell, but instead adds higher-order correction terms through a successive differentiation. As a result, the existing code structure and solution procedure can be maintained, and the high-order correction terms can be modularized. Verification cases using simple grids against analytical solutions demonstrate that the developed 4th order scheme can provide up to 5th order accuracy. Validation studies illustrate significant reduction in grid requirements using the high-order scheme in resolving near-wall turbulence down to the Kolmogorov scale. The present high-order method adds 20-50% CPU overhead in solving 3D problems, and can be applied to an existing CFD solver with minimal modification.