一种无积分任意四边形非结构化网格节点间断Galerkin方法  

A Quadrature-Free Scheme for Nodal Discontinuous Galerkin Method on Arbitrary Quadrilateral Unstructured Meshes

作　　者：李龙翔 张庆河[1] Li Longxiang;Zhang Qinghe(State Key Laboratory of Hydraulic Engineering Simulation and Safety,Tianjin University,Tianjin 300072,Chin)

机构地区：[1]天津大学水利工程仿真与安全国家重点实验室,天津300072

出　　处：《天津大学学报（自然科学与工程技术版）》2018年第6期575-582,共8页Journal of Tianjin University：Science and Technology

基　　金：国家重点研发计划资助项目(2017YFC1404204)~~

摘　　要：节点间断Galerkin方法是近年来得到迅速发展的高精度数值方法,可以采用任意多边形网格对平面求解域进行离散.针对任意四边形非结构化网格,传统的节点间断Galerkin方法采用数值积分对离散方程进行计算,需要较大的计算量与存储空间.为了提高任意四边形非结构化网格上节点间断Galerkin方法的计算效率,提出了一种新的无积分格式实现方法,即将积分节点与插值节点定义为同一节点集,并利用节点基函数的插值性质,推导出每个单元内控制方程的无积分离散格式.通过在任意四边形非结构化网格中对二维对流方程进行数值求解,验证了新提出的无积分方法的准确性和计算效率.结果表明,无积分方法与传统数值积分方法计算误差和收敛精度基本相同,而其计算效率提高1倍以上.The nodal discontinuous Galerkin method is one of high order numerical methods,which has rapidly devel-oped in recent years,and through which plane solution domain can be discretized in arbitrary polygonal meshes. For arbitrary quadrilateral unstructured meshes,the traditional discontinuous Galerkin method employs Gauss quadrature formula to compute the discrete control equations,which requires a very large computational cost and memory. To im-prove the computational efficiency for the nodal discontinuous Galerkin method on arbitrary quadrilateral unstructuredmeshes,a new quadrature-free scheme was proposed. Specifically,the new scheme defines integral and interpolation nodes as the same node set,and utilizes the interpolation property of nodal basis functions to discretize the control equa-tions into a quadrature-free formula in each element. Through solving the two-dimensional advection problem on arbi-trary quadrilateral unstrustured meshes,the accuracy and efficiency of the new quadrature-free scheme are verified. The results indicate that despite similar numerical error and convergence rate,the new quadrature-free scheme shows highercomputational efficiency,at least twice that of the traditional numerical quadrature scheme.

关 键 词：节点间断Galerkin方法 任意四边形单元 二维对流扩散方程 

分 类 号：O242[理学—计算数学]