Controlling conservation laws II: Compressible Navier–Stokes equations

Abstract

We propose, study, and compute solutions to a class of optimal control problems for hyperbolic systems of conservation laws and their viscous regularization [17]. 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.