Controlling conservation laws II: compressible Navier-Stokes equations

TL;DR

This paper develops a novel optimal control framework for the compressible Navier-Stokes equations, introducing a primal-dual variational approach and an implicit, stable numerical scheme validated through numerical experiments.

Contribution

It introduces a new variational formulation and a primal-dual algorithm for controlling compressible Navier-Stokes equations, with a focus on stability and numerical implementation.

Findings

The proposed algorithm is unconditionally stable.

Numerical examples demonstrate the effectiveness of the control method.

The primal-dual approach successfully solves the variational problem for BNS.

Abstract

We propose, study, and compute solutions to a class of optimal control problems for hyperbolic systems of conservation laws and their viscous regularization. We take barotropic compressible Navier--Stokes equations (BNS) as a canonical example. We first apply the entropy--entropy flux--metric condition for BNS. We select an entropy function and rewrite BNS to a summation of flux and metric gradient of entropy. We then develop a metric variational problem for BNS, whose critical points form a primal-dual BNS system. We design a finite difference scheme for the variational system. The numerical approximations of conservation laws are implicit in time. We solve the variational problem with an algorithm inspired by the primal-dual hybrid gradient method. This includes a new method for solving implicit time approximations for conservation laws, which seems to be unconditionally stable. Several numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.