Spectral Difference Method with a Posteriori Limiting: III- Navier-Stokes Equations with Arbitrary High-Order Accuracy

TL;DR

This paper introduces a high-order spectral difference method with a posteriori limiting for Navier-Stokes equations, capable of accurately capturing shocks and dissipative scales at lower resolutions with exponential convergence for smooth solutions.

Contribution

It integrates an arbitrarily high-order Laplacian operator into the spectral difference method with shock-capturing capabilities, enhancing accuracy and efficiency.

Findings

The method achieves exponential convergence on smooth solutions.

It effectively captures shocks and dissipative scales at lower resolutions.

The approach recovers high-order solutions for dissipative scales.

Abstract

We incorporate an arbitrarily high-order method for the Laplacian operator into the Spectral Difference method (SD). The resulting method is capable of capturing shocks thanks to its a-posteriori limiting methodology, and therefore it is able to survive scenarios in which the dissipative scales (viscous and diffusive) are not properly described. Moreover, it is capable of capturing these scales at lower resolution compared to lower-order methods and therefore attains convergence at lower resolution. We show that the method at hand has exponential convergence when describing smooth solutions and is able to recover a high-order solution when solving the dissipative scales.