The aim of the present paper is to provide a comparison between several finite-volume methods of different numerical accuracy: the second-order Godunov method with PPM interpolation and the high-order finite-volume WENO method. The results show that while on a smooth problem the high-order method performs 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 at cell interfaces. Most important, results in terms of turbulent spectra are similar regardless of the numerical methods employed, except that the PPM method fails to provide an accurate representation in the high-frequency range of the spectra. 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.
"Investigation of finite-volume methods to capture shocks and turbulence spectra in compressible flows." Commun. Appl. Math. Comput. Sci. 15 (1) 1 - 36, 2020. https://doi.org/10.2140/camcos.2020.15.1_