An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary
An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary
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边界风向变化双曲守恒定律的逆 Lax-Wendroff 过程
DOI：
10.1016/j.jcp.2020.109940
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发表时间：
2020-10
影响因子：
4.1
通讯作者：
Mengping Zhang
中科院分区：
文献类型：
--
作者：
Jianfang Lu;Chi-Wang Shu;Sirui Tan;Mengping Zhang
在本文中，我们重新考虑逆Lax-Wendroff（ILW）程序，这是一个数值边界处理求解双曲型守恒律，并提出了一种新的方法来评估鬼点上的值。ILW过程首先被提出来处理当物理边界与笛卡尔网格以任意方式相交时的“切割单元”问题。ILW程序的核心思想是反复利用偏微分方程（PDE）和流入边界条件，以获得边界上的各阶法向导数。在[28]中提出了一种简化的ILW程序，并仅将ILW程序用于一阶法向导数的评估。简化ILW方法和本文提出的ILW方法之间的主要区别在于，我们在鬼点上分别定义了未知量u和通量f（u）。这种处理的一个优点是它允许雅可比矩阵f′（u）的特征值在边界上接近于零，这可能出现在许多物理问题中。我们还提出了一种新的加权基本无振荡（韦诺）型外流边界外推，其思想来自于文[32]中的多分辨率韦诺格式。如果解在边界附近是光滑的，韦诺型外推保持高阶精度;如果激波靠近边界，WENO型外推自动变为低阶外推。这种韦诺型外推保持了自相似性，因而在计算双曲型守恒律时更有优势。我们提供了大量的数值例子来证明我们的方法是稳定的，高阶精度和具有不同类型的边界条件，包括固体壁边界条件，当物理边界与网格不对齐的各种问题具有良好的性能。
In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f (u) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f′(u) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32]. The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.
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