Capturing shocks and turbulence spectra in compressible flows. Part 1: Comparison of low and high-order finite-volume methods.

The aim of the Part 1 of the present paper is to provide a comparison between several finite-volume methods: second-order Godunov method with PPM interpolation and high-order finite-volume WENO methods with several popular reconstruction variants. The results show that while on a smooth problem the high-order methods perform better than the second-order one, when the solution contains a shock all the methods collapse to first-order accuracy. In the context of the decay of compressible homogeneous isotropic turbulence with shocklets, the actual overall order of accuracy of the methods reduces to second-order, despite the use of fifth-order reconstruction schemes. Most important, results in terms of turbulent spectra are similar regardless of the numerical methods employed. However it is shown that the PPM method fails to provide an accurate representation in the high-frequency range of the spectra. In the Part 2 of the present paper, it is found that this specific issue comes from the slope-limiting procedure and a novel hybrid PPM/WENO method is developed that has the ability to capture the turbulent spectra with the accuracy of a high-order method, but at the cost of the second-order Godunov method. Overall, it is shown that virtually the same physical solution can be obtained much faster by refining a simulation with the second-order method and carefully chosen numerical procedures, rather than running a coarse high-order simulation. Our results demonstrate the importance of evaluating the accuracy of a numerical method in terms of its actual spectral dissipation and dispersion properties on mixed smooth/shock cases, rather than by the theoretical formal order of convergence rate.