We present an optimal transport approach for mesh adaptivity and shock capturing of compressible flows. Shock capturing is based on a viscosity regularization of the governing equations by introducing an artificial viscosity field as solution of the Helmholtz equation. Mesh adaptation is based on the optimal transport theory by formulating a mesh mapping as solution of Monge-Ampere equation. The marriage of optimal transport and viscosity regularization for compressible flows leads to a coupled system of the compressible Euler/Navier-Stokes equations, the Helmholtz equation, and the Monge-Ampere equation. We propose an iterative procedure to solve the coupled system in a sequential fashion using homotopy continuation to minimize the amount of artificial viscosity while enforcing positivity-preserving and smoothness constraints on the numerical solution. We explore various mesh monitor functions for computing r-adaptive meshes in order to reduce the amount of artificial dissipation and improve the accuracy of the numerical solution. The hybridizable discontinuous Galerkin method is used for the spatial discretization of the governing equations to obtain high-order accurate solutions. Extensive numerical results are presented to demonstrate the optimal transport approach on transonic, supersonic, hypersonic flows in two dimensions. The approach is found to yield accurate, sharp yet smooth solutions within a few mesh adaptation iterations.

翻译：我们提出了一种基于最优传输的可压缩流网格自适应与激波捕捉方法。激波捕捉通过在控制方程中引入由亥姆霍兹方程求解得到的人工黏性场实现黏性正则化。网格自适应则基于最优传输理论，将网格映射构建为蒙日-安培方程的解。将最优传输与黏性正则化相结合处理可压缩流，形成了由可压缩欧拉/纳维-斯托克斯方程、亥姆霍兹方程和蒙日-安培方程构成的耦合系统。我们提出了一种迭代求解策略，采用同伦延拓法顺序求解该耦合系统，在确保数值解保正性与光滑性约束的同时，最大限度降低人工黏性量。为减少人工耗散并提高数值解精度，我们探索了多种用于计算r-自适应网格的监测函数。采用混合化间断伽辽金方法对控制方程进行空间离散，以获得高阶精度数值解。通过二维跨声速、超声速和高超声速流动的大量数值算例，验证了该最优传输方法的效果。结果表明该方法仅需少量网格自适应迭代即可获得精确、清晰且光滑的数值解。