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.

中文翻译：

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rp - 可压缩流动的适应

我们提出了一种 $rp$-adaptation 策略，用于对具有冲击的可压缩无粘性流动进行高保真模拟。流动不连续区域的网格分辨率通过使用变分优化器来增加，以适应那里的网格和集群自由度。在平滑流动的区域，我们分别通过增加或减少元素的多项式阶数来局部增加或减少局部分辨率。这种双重方法使我们能够利用两种方法的优势来获得最佳计算性能，从而降低模拟的总体成本。适应工作流使用传感器来计算不连续性和平滑区域，计算成本低，但该框架是通用的，可以与其他基于特征的传感器或误差估计器结合使用。我们在跨音速和超音速流态下使用两种几何形状来演示这种概念验证。该方法已在开源光谱/$hp$ 元素框架$Nektar++$ 及其专用的高阶网格生成工具$NekMesh$ 中实现。结果表明，建议的 $rp$-adaptation 方法是一种提高准确性的合理成本有效的方法。