Optimal transport for mesh adaptivity and shock capturing of compressible flows

TL;DR

This paper introduces a novel optimal transport-based method for mesh adaptivity and shock capturing in compressible flows, combining viscosity regularization, Monge-Ampere equations, and high-order discretization for accurate solutions.

Contribution

It develops a coupled system integrating optimal transport, viscosity regularization, and high-order discretization, with an iterative solution procedure for efficient shock capturing and mesh adaptation.

Findings

Accurate, sharp, and smooth solutions achieved in transonic to hypersonic flows.

Few mesh adaptation iterations needed for high-quality solutions.

Method reduces artificial dissipation and improves numerical accuracy.

Abstract

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.