A posteriori error estimation for the Lax-Wendroff finite difference scheme
A posteriori error estimation for the Lax-Wendroff finite difference scheme
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Lax-Wendroff 有限差分格式的后验误差估计
DOI：
10.1016/j.cam.2013.12.035
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
S. Tavener
中科院分区：
文献类型：
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作者：
J. B. Collins;D. Estep;S. Tavener
在许多应用领域中，求解双曲型偏微分方程（如守恒律）的首选方法是采用有限差分格式。虽然有限差分格式是服从物理解释，有限差分公式的一个缺点是，它是相对困难的，以获得所谓的面向目标的后验误差估计。后验误差估计提供了一种计算方法，以数值方式计算从数值解计算的指定量的误差的精确估计。后验误差估计在数值模拟的可靠性量化和有效的自适应误差控制等方面有着广泛的应用，但有限差分格式后验误差估计的主要困难在于引入变分公式和伴随问题以及残差的系统定义。在本文中，我们处理这个问题，首先推导出一个等价的有限元方法和Lax-Wendroff有限体积法。然后，我们得到了一个伴随的误差表示公式的解决方案与这种方法。线性和非线性粘性守恒律的结果。
In many application domains, the preferred approaches to the numerical solution of hyperbolic partial differential equations such as conservation laws are formulated as finite difference schemes. While finite difference schemes are amenable to physical interpretation, one disadvantage of finite difference formulations is that it is relatively difficult to derive the so-called goal oriented a posteriori error estimates. A posteriori error estimates provide a computational approach to numerically compute accurate estimates in the error in specified quantities computed from a numerical solution. Widely used for finite element approximations, a posteriori error estimates yield substantial benefits in terms of quantifying reliability of numerical simulations and efficient adaptive error control.The chief difficulties in formulating a posteriori error estimates for finite difference schemes is introducing a variational formulation–and the associated adjoint problem–and a systematic definition of residual errors. In this paper, we approach this problem by first deriving an equivalency between a finite element method and the Lax–Wendroff finite volume method. We then obtain an adjoint based error representation formula for solutions obtained with this method. Results from linear and nonlinear viscous conservation laws are given.