论文标题
控制保护法II：可压缩的Navier-Stokes方程
Controlling conservation laws II: compressible Navier-Stokes equations
论文作者
论文摘要
我们建议，研究和计算解决方案的解决方案，以解决一系列最佳控制问题，以解决保护法的双曲线系统及其粘性正则化。我们以压缩式Navier（BNS）为典型示例。我们首先为BN应用熵 - 内向通量条件。我们选择一个熵函数，然后将BNS重写为熵的通量和度量梯度的总和。然后，我们为BNS开发一个度量变异问题，BNS的临界点形成了原始的双重BNS系统。我们为变分系统设计有限的差异方案。保护定律的数值近似是隐含的。我们通过受原始二重混合梯度方法启发的算法解决了变异问题。这包括一种新方法，用于解决保护定律的隐式时间近似，这似乎是无条件稳定的。提出了几个数值示例，以证明所提出的算法的有效性。
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.
