Controlling conservation laws II: Compressible Navier–Stokes equations

• Published in: Journal of computational physics 2022-08, Vol.463, p.111264, Article 111264
• Main Authors: Li, Wuchen, Liu, Siting, Osher, Stanley
• Format: Article
• Language: English
• Subjects: Algorithms; Compressibility; Computational physics; Conservation laws; Critical point; Entropy; Entropy–entropy flux–metric; Finite difference method; Fisher information; Fluid flow; Hyperbolic systems; Lax—Friedrichs scheme; Navier-Stokes equations; Optimal control; Primal–dual algorithm; Regularization
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.111264

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. •We propose and compute solutions of barotropic compressible Navier–Stokes equations.•We develop an entropy–entropy flux–metric condition for barotropic compressible Navier–Stokes equations. Using it, we construct a metric variational problem.•We design variational implicit schemes for regularized conservation law systems. The variational scheme is solved by a primal–dual hybrid gradient method.