控制守恒律II：可压缩Navier-斯托克斯方程
Controlling conservation laws II: compressible Navier-Stokes equations
摘要：
我们提出，研究，并计算解决方案的一类最优控制问题的双曲型守恒律及其粘性正则化。本文以正压可压缩Navier-斯托克斯方程（BNS）为典型算例。我们首先对BNS应用熵--熵通量--度量条件。我们选择一个熵函数，将BNS改写为熵的通量和度量梯度之和。然后，我们发展了一个度量变分问题的BNS，其临界点形成一个原始-对偶BNS系统。我们设计了一个有限差分格式的变分系统。守恒律的数值逼近在时间上是隐式的。我们解决的变分问题的算法启发的原始-对偶混合梯度法。这包括一个新的方法来解决隐式时间近似守恒律，这似乎是无条件稳定的。数值算例验证了算法的有效性。
英文摘要：
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.
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