标题： 基于隐式激波追踪的精确分辨率激波主导粘性流动的$rp$自适应方法 显示英文标题

标题： An $rp$-adaptive method for accurate resolution of shock-dominated viscous flow based on implicit shock tracking

摘要： 本文介绍了一种基于优化的$rp$-自适应数值方法，用于使用隐式激波追踪和高阶间断伽辽金离散化在传统粗网格上近似粘性、激波主导流场的解，而无需非线性稳定（例如人工黏性或限制器）。 所提出的方法对最初为使网格面与解间断对齐而开发的隐式激波追踪方法进行了改进，将其压缩为粘性激波和边界层，作为一种新颖的激进$r$- 自适应方法。 这种形式的$r$- 自适应自然地通过网格节点的离散流动变量和坐标上的富集残差最小化来实现。 为了确保在粘性特征处有足够的网格压缩以避免稳定性的需求，提出了对激波追踪优化求解器的几种创新方法，包括残差加权、步长约束和修改以及基于粘度的连续性。 最后，$p$- 自适应被用来局部增加多项式阶数，具有三个明显的好处：（1）减少了激波波前和边界层附近网格压缩的需求；（2）减少了在给定网格拓扑结构下$r$- 自适应不足以覆盖的区域的误差；（3）通过在最粗网格上执行大部分$r$- 自适应迭代来降低计算成本。 一系列数值实验表明，所提出的方法能够有效地解决粘性、激波主导的流场问题，包括准确预测由超音速流过圆柱体产生的热流分布，并且在每自由度精度方面与高阶离散化的$h$- 自适应方法相比表现良好。 显示英文摘要

摘要： This work introduces an optimization-based $rp$-adaptive numerical method to approximate solutions of viscous, shock-dominated flows using implicit shock tracking and a high-order discontinuous Galerkin discretization on traditionally coarse grids without nonlinear stabilization (e.g., artificial viscosity or limiting). The proposed method adapts implicit shock tracking methods, originally developed to align mesh faces with solution discontinuities, to compress elements into viscous shocks and boundary layers, functioning as a novel approach to aggressive $r$-adaptation. This form of $r$-adaptation is achieved naturally as the minimizer of the enriched residual with respect to the discrete flow variables and coordinates of the nodes of the grid. Several innovations to the shock tracking optimization solver are proposed to ensure sufficient mesh compression at viscous features to render stabilization unnecessary, including residual weighting, step constraints and modifications, and viscosity-based continuation. Finally, $p$-adaptivity is used to locally increase the polynomial degree with three clear benefits: (1) lessens the mesh compression requirements near shock waves and boundary layers, (2) reduces the error in regions where $r$-adaptivity is not sufficient with the given grid topology, and (3) reduces computational cost by performing a majority of the $r$-adaptivity iterations on the coarsest discretization. A series of numerical experiments show the proposed method effectively resolves viscous, shock-dominated flows, including accurate prediction of heat flux profiles produced by hypersonic flow over a cylinder, and compares favorably in terms of accuracy per degree of freedom to $h$-adaptation with a high-order discretization.