Assessment of time implicit discretizations for the computation of turbulent compressible flows

Abstract

Restrictions on the maximum allowable time step of explicit time integration methods for direct and large eddy simulations of compressible turbulent flows at high Reynolds numbers can be very severe, because of the extremely small space steps used close to solid walls to capture tiny and elongated boundary layer structures. A way of increasing stability limits is to use implicit time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. A successful implicit time scheme for scale-resolving simulations should provide the best possible compromise between these opposite requirements. In this paper, several implicit schemes are assessed against two explicit time integration techniques, namely a standard four-stage and a six-stage optimized Runge–Kutta method, in terms of computational cost required to achieve a threshold level of accuracy. Precisely, a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations is compared to time-accurate Runge–Kutta implicit residual smoothing (IRS) schemes. A new IRS scheme of fourth-order accuracy, based on a bilaplacian operator, is developed to improve the accuracy of the classical second-order approach. Numerical results show that the proposed IRS scheme leads to reductions in computational time by about a factor 5 for an accuracy comparable to that of the corresponding explicit Runge-Kutta scheme.