Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical $L^2$ projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.

由于经典的WENO方案[27]可能会遭受轻微的震荡后振荡（这是导致数值残差悬挂在截断误差水平上的原因），因此新的高阶多分辨率WENO方案[59]得以成功解决对于稳态问题，我们将这些高阶有限体积多分辨率WENO技术用作模拟稳态问题的高阶Runge-Kutta间断Galerkin（RKDG）方法的限制器。首先，设计了一种新的故障电池指示器，以精确检测需要进一步限制程序的电池。然后在一个层次序列上采用高阶多分辨率WENO限制程序${\mathrm{大号}}^{\mathrm{2个}}$故障单元本身内DG解决方案的投影多项式。这样做，这些具有多分辨率WENO限制器的RKDG方法可以从最佳的高阶精度逐渐降低到强不连续点附近的一阶精度，抑制轻微的震荡后振荡，并推动数值残差稳定下来。在稳态仿真中将机器置零。这些新型的多分辨率WENO限制器非常容易构造，并且可以轻松实现任意高阶精度，以解决多维稳态问题。