$rp$-adaptation for compressible flows

TL;DR

This paper introduces an $rp$-adaptation strategy combining mesh refinement and polynomial order adjustment for efficient high-fidelity simulations of compressible flows with shocks, demonstrated on transonic and supersonic cases.

Contribution

The paper presents a novel dual $rp$-adaptation method that optimally combines mesh refinement and polynomial order variation for compressible flow simulations.

Findings

Effective reduction in computational cost while maintaining accuracy.

Successful implementation in open-source spectral/$hp$ element framework Nektar++.

Applicable to transonic and supersonic flow regimes.

Abstract

We present an $rp$-adaptation strategy for high-fidelity simulation of compressible inviscid flows with shocks. The mesh resolution in regions of flow discontinuities is increased by using a variational optimiser to $r$-adapt the mesh and cluster degrees of freedom there. In regions of smooth flow, we locally increase or decrease the local resolution through increasing or decreasing the polynomial order of the elements, respectively. This dual approach allows us to take advantage of the strengths of both methods for best computational performance, thereby reducing the overall cost of the simulation. The adaptation workflow uses a sensor for both discontinuities and smooth regions that is cheap to calculate, but the framework is general and could be used in conjunction with other feature-based sensors or error estimators. We demonstrate this proof-of-concept using two geometries at transonic and supersonic flow regimes. The method has been implemented in the open-source spectral/$hp$ element framework $Nektar++$, and its dedicated high-order mesh generation tool $NekMesh$. The results show that the proposed $rp$-adaptation methodology is a reasonably cost-effective way of improving accuracy.