The numerical solutions to nonlinear hyperbolic balance laws at (or near) steady state may develop spurious oscillations due to the imbalance between flux and source terms. In the present article, we study a high order well-balanced discontinuous Galerkin (DG) scheme for balance law with subsonic flow, which preserves equilibrium solutions of the flow exactly, and also provides non-oscillatory solutions for flow near equilibrium. The key technique is to reformulate the DG scheme in terms of global equilibrium variables which remain constant in space and time, and are obtained by rewriting the balance law in conservative form. We show that the proposed scheme is well-balanced and validate the scheme for various flows given by $2\times 2$ hyperbolic balance law. We also extend the scheme to flows on networks, particularly to include the coupling conditions at nodes of the network.

由于通量和源项之间的不平衡，稳态（或接近稳态）下非线性双曲平衡定律的数值解可能会产生伪振荡。在本文中，我们研究了具有亚音速流的平衡律的高阶平衡良好的不连续伽勒金（DG）方案，该方案精确地保留了流的平衡解，并且还为平衡附近的流提供了非振荡解。关键技术是根据全局平衡变量来重新构造DG方案，该全局平衡变量在空间和时间上保持不变，并且是通过以保守形式重写平衡定律而获得的。我们证明了所提出的方案是均衡的，并针对由$2\times 2$双曲平衡定律。我们还将方案扩展到网络上的流，特别是包括网络节点上的耦合条件。