Weights Design For Maximal Order WENO Schemes
Weights Design For Maximal Order WENO Schemes
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DOI：
10.1007/s10915-013-9810-0
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发表时间：
2013-12
影响因子：
2.5
通讯作者：
F. Aràndiga;M. C. Martí;P. Mulet
中科院分区：
文献类型：
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作者：
F. Aràndiga;M. C. Martí;P. Mulet
加权基本非振荡（WENO）有限差分格式，由Liu等人（computational Phys 115(1):200 - 212,1994）提出，由Jiang和Shu （computational Phys 126(1):202 - 22,1996）改进，是最常用的近似双曲方程解的方法之一。但这些方案在光滑极值附近不能提供最大阶精度，此时解的一阶导数为零。一些作者用不同的重量设计来解决这个问题。在本文中，我们主要关注Yamaleev和Carpenter提出的权重（J computer Phys 228:4248-4272, 2009）。他们提出了新的权重，以提供比Borges等人提出的更快的权重收敛（J Comput Phys 227:3191 - 3211,2008），并推导了一些对权重参数的约束，以保证WENO方案对于具有任意数量的消失导数的充分光滑解具有最大阶。我们用Yamaleev和Carpenter （J computer Phys 228:4248-4272, 2009）提出的权值对该方案进行了分析，并证明了该方案在不连续点附近的阶数比经典WENO方案差。为了解决这些精度问题，我们在Yamaleev和Carpenter （J computer Phys 228:4248-4272, 2009）提出的基础上定义了新的权值，并对权值参数进行了约束，以保证所得到的格式的最大阶精度。
Weighted essentially non-oscillatory (WENO) finite difference schemes, developed by Liu et al. (Comput Phys 115(1):200–212, 1994) and improved by Jiang and Shu (Comput Phys 126(1):202–228, 1996), are one of the most popular methods to approximate the solutions of hyperbolic equations. But these schemes fail to provide maximal order accuracy near smooth extrema, where the first derivative of the solution becomes zero. Some authors have addressed this problem with different weight designs. In this paper we focus on the weights proposed by Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009). They propose new weights to provide faster weight convergence than those presented in Borges et al. (J Comput Phys 227:3191–3211, 2008) and deduce some constraints on the weights parameters to guarantee that the WENO scheme has maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. We analyze the scheme with the weights proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009) and prove that near discontinuities it achieves worse orders than classical WENO schemes. In order to solve these accuracy problems, we define new weights, based on those proposed in Yamaleev and Carpenter (J Comput Phys 228:4248–4272, 2009), and get some constraints on the weights parameters to guarantee maximal order accuracy for the resulting schemes.