Discontinuous Galerkin Methods for Nonlinear Scalar Hyperbolic Conservation Laws:Divided Difference Estimates and Accuracy Enhancement

浙江大学数学科学学院九十周年院庆系列活动之五

题目：Discontinuous Galerkin Methods for Nonlinear Scalar Hyperbolic Conservation Laws:Divided Difference Estimates and Accuracy Enhancement

报告人：孟雄(哈尔滨工业大学数学系副教授)

时间：6月6日 下午15:30―16:30

地点：逸夫工商楼200―9

报告人简介：哈尔滨工业大学数学系青年拔尖人才、副教授、硕导，欧盟玛丽居里学者，主要研究方向为计算流体力学的高阶精度数值方法，包括有限差分和有限体积的加权本质无振荡(WENO)方法、间断有限元(DG)方法等。

摘要：In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalarnonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the $alpha$-th order $(1 le alpha le k+1)$ divided difference of the DG error in the$L^2$ norm is of order $k+3/2-alpha/2$ when upwind fluxes are used, under the condition that$|'(u)$ possesses a uniform positive lower bound. By the duality argument,we then derive superconvergence results of order $2k+3/2-alpha/2$ in the negative-order norm, demonstrating that it is possibleto extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least$3k/2+1$th order superconvergence for post-processed solutions. As a by-product,for variable coefficient hyperbolic equations, we provide an explicit proof foroptimal convergence results of order $k+1$ in the $L^2$ norm for the divided differences of DG errors and thus $(2k+1)$th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results. 

联系人：仲杏慧（zhongxh@zju.edu.cn）

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