论文标题
浅水方程的均衡级别五阶有限差异HERMITE WENO方案
Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations
论文作者
论文摘要
在本文中，我们提出了一个均衡平衡的五阶有限差异hemite weno（Hweno）方案，该方程的浅水方程为非平衡底部形状，以均衡形式。为了实现井平衡属性，我们采用了类似的WENO-XS方案思想[Xing and Shu，J。Comput。 Phys。，208（2005），206-227。]平衡通量梯度和源术语。原始方程中的通量由非线性Hweno重建重建，而衍生方程中的其他磁通直接由高度多项式近似。在时间离散步骤中，将HWENO限制器应用于平衡变量的衍生物，以控制维持良好平衡特性的虚假振荡。所提出的Hweno方案不用在相同的五阶Weno-XS方案中使用五点模具，而只需要重建中的紧凑型三分模板即可。提出了一个和二维中的各种基准示例，以表明HWENO方案是五阶准确度，保留稳态溶液，具有更好的分辨率，更准确，更有效，并且本质上是非振荡的。
In this paper, we propose a well-balanced fifth-order finite difference Hermite WENO (HWENO) scheme for the shallow water equations with non-flat bottom topography in pre-balanced form. For achieving the well-balance property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput. Phys., 208 (2005), 206-227.] to balance the flux gradients and the source terms. The fluxes in the original equation are reconstructed by the nonlinear HWENO reconstructions while other fluxes in the derivative equations are approximated by the high-degree polynomials directly. And an HWENO limiter is applied for the derivatives of equilibrium variables in time discretization step to control spurious oscillations which maintains the well-balance property. Instead of using a five-point stencil in the same fifth-order WENO-XS scheme, the proposed HWENO scheme only needs a compact three-point stencil in the reconstruction. Various benchmark examples in one and two dimensions are presented to show the HWENO scheme is fifth-order accuracy, preserves steady-state solution, has better resolution, is more accurate and efficient, and is essentially non-oscillatory.
