The solution of conservation laws with parametrized shock waves presents challenges for both high-order numerical methods and model reduction techniques. We introduce an r-adaptivity scheme based on optimal transport and apply it to develop reduced order models for compressible flows. The optimal transport theory allows us to compute high-order r-adaptive meshes from a starting reference mesh by solving the Monge-Ampere equation. A high-order discretization of the conservation laws enables high-order solutions to be computed on the resulting r-adaptive meshes. Furthermore, the Monge-Ampere solutions contain mappings that are used to reduce the spatial locality of the resulting solutions and make them more amenable to model reduction. We use a non-intrusive model reduction method to construct reduced order models of both the mesh and the solution. The procedure is demonstrated on three supersonic and hypersonic test cases, with the hybridizable discontinuous Galerkin method being used as the full order model.

翻译：具有参数化激波间断的守恒律求解对高阶数值方法与模型降阶技术均构成挑战。本文提出一种基于最优输运的r-自适应方案，并将其应用于构建可压缩流降阶模型。通过求解Monge-Ampere方程，最优输运理论允许我们从初始参考网格计算高阶r-自适应网格。守恒律的高阶离散化使得在所得r-自适应网格上可计算高阶解。此外，Monge-Ampere解包含的映射关系可用于降低所得解的空间局域性，使其更易于进行模型降阶。我们采用非侵入式模型降阶方法构建网格与解两者的降阶模型。该流程通过三个超声速与高超声速算例进行验证，其中全阶模型采用可杂交间断伽辽金方法。