# Capturing shocks and turbulence spectra in compressible flows. Part 2: A new hybrid PPM/WENO method

## TL;DR

This paper introduces a hybrid PPM/WENO method that accurately captures turbulence spectra in compressible flows at lower computational cost, improving upon traditional methods by addressing high-frequency spectral inaccuracies.

## Contribution

A novel hybrid PPM/WENO scheme is developed, combining high-order accuracy with low computational cost for turbulence spectrum capture in compressible flows.

## Key findings

- The hybrid method matches high-order schemes in spectral accuracy.
- It achieves similar results to high-order methods at second-order computational cost.
- Refined second-order simulations can outperform coarse high-order ones in efficiency.

## Abstract

In the Part 1 of the present paper the performance of several different low and high-order finite-volume methods were assessed by investigating how well they can capture the turbulent spectra of a compressible flow where small smooth turbulent structures interact with shocks and discontinuities. The comparisons showed that a second-order Godunov method with PPM interpolation provides results virtually the same as a fourth-order WENO scheme but at a significant lower cost. However, it is shown that the PPM method fails to provide an accurate representation in the high-frequency range of the spectra. In the present paper we show that this specific issue comes from the slope-limiting procedure and a novel hybrid PPM/WENO method is developed, which has the ability to capture the turbulent spectra with the accuracy of a formally 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.

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Source: https://tomesphere.com/paper/1902.06666