求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法

doi:10.19596/j.cnki.1001-246x.7919

计算物理 ›› 2019, Vol. 36 ›› Issue (5): 517-532.DOI: 10.19596/j.cnki.1001-246x.7919

求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法

赵国忠¹, 蔚喜军², 郭虹平¹, 董自明¹

1. 包头师范学院数学科学学院, 包头 014030;
2. 北京应用物理与计算数学研究所计算物理实验室, 北京 100088

收稿日期:2018-07-04 修回日期:2018-09-22 出版日期:2019-09-25 发布日期:2019-09-25
通讯作者: Liu Haifeng (1968-),condensed matter physics,E-mail:liu_haifeng@iapcm.ac.cn
作者简介:Zhao Guozhong (1977-),male,PhD,professor,research in computational fluid dynamics,E-mail:zhaoguozhongbttc@sina.com
基金资助:
National Natural Science Foundation of China (11761054, 11571002, 11261035), Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-15-A07), Natural Science Foundation of Inner Mongolia Autonomous Region, China (2015MS0108, 2012MS0102) and Science Research Foundation of Institute of Higher Education of Inner Mongolia Autonomous Region, China (NJZZ12198, NJZZ16234, NJZZ16235)

A Local Discontinuous Petrov-Galerkin Method for Partial Differential Equations with High Order Derivatives

ZHAO Guozhong¹, YU Xijun², GUO Hongping¹, DONG Ziming¹

1. Faculty of Mathematics, Baotou Teachers'College, Baotou 014030, China;
2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Received:2018-07-04 Revised:2018-09-22 Online:2019-09-25 Published:2019-09-25
Supported by:
National Natural Science Foundation of China (11761054, 11571002, 11261035), Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT-15-A07), Natural Science Foundation of Inner Mongolia Autonomous Region, China (2015MS0108, 2012MS0102) and Science Research Foundation of Institute of Higher Education of Inner Mongolia Autonomous Region, China (NJZZ12198, NJZZ16234, NJZZ16235)

摘要/Abstract

摘要： 构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数，包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组，再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用，如孤立子的传播及相互碰撞等.

关键词: KdV型方程, 双调和方程, 局部间断Petrov-Galerkin方法, 孤立子演化

Abstract: A local discontinuous Petrov-Galerkin method is proposed for solving three types of partial differential equations with second, third and fourth order derivatives, respectively. They are Burgers type equations, KdV type equations and bi-harmonic type equations. The method extends discontinuous Petrov-Galerkin method for conservation laws by rewriting corresponding equations into a first order system and solving the system instead of the original equation. The method has a fourth order accuracy and maintains advantages of discontinuous Petrov-Galerkin method. Numerical simulations verify that the method reaches optimal convergence order and simulates well complex wave interaction such as soliton propagation and collision.

Key words: KdV type equation, bi-harmonic equation, local discontinuous Petrov-Galerkin method, soliton evolution

中图分类号:

O241.82

赵国忠, 蔚喜军, 郭虹平, 董自明. 求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法[J]. 计算物理, 2019, 36(5): 517-532.

ZHAO Guozhong, YU Xijun, GUO Hongping, DONG Ziming. A Local Discontinuous Petrov-Galerkin Method for Partial Differential Equations with High Order Derivatives[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2019, 36(5): 517-532.

参考文献

[1] SHEN J. A new dual-Petrov-Galerkin method for third and higher odd-order differential equations:Application to the KdV equation[J]. SIAM J Numer Anal, 2003, 41(5):1595-1619.
[2] LIU H L, YAN J. A local discontinuous Galerkin method for the Korteweg-de Vries equation with boundary effect[J]. J Comput Phys, 2006, 215:197-218.
[3] YI N Y, HUANG Y Q. A direct discontinuous Galerkin method for the generalized Korteweg-de Vries equation:Energy conservation and boundary effect[J]. J Comput Phys, 2013, 242:351-366.
[4] MA H, SUN W. A Legendre-Petrov-Galerkin method and Chebyshev collocation method for the third-order differential equations[J]. SIAM J Numer Anal, 2000, 38:1425-1438.
[5] MA H, SUN W. Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation[J]. SIAM J Numer Anal, 2001, 39:1380-1394.
[6] LUO D M, HUANG W Z, QIU J X. A hybrid LDG-HWENO scheme for KdV-type quations[J]. J Comput Phys, 2016, 313:754-774.
[7] DEBUDDCHE A, PRINTEMS J. Numerical simulation of the stochastic Korteweg-de Vries equation[J]. Phys D, 1999, 134:200-226.
[8] XU Y, Shu C W. Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations[J]. Comput Method Appl M, 2006, 195:3430-3447.
[9] REED W H, HILL T R. Triangular mesh methods for the neutron transport equation[R]. Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973.
[10] COCKBURN B, SHU C W. The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws[J]. Math Model Numer Anal (M2AN), 1991, 25(3):337-361.
[11] COCKBURN B, SHU C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II:General framework[J]. Math Comp, 1999, 52(186):411-435.
[12] COCKBURN B, LIN S Y, SHU C W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III:One-dimensional systems[J]. J Comput Phys, 1989, 84(1):90-113.
[13] COCKBURN B, HOU S, SHU C W. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV:The multidimensional case[J]. Math Comp, 1990, 54(190):545-581.
[14] COCKBURN B, SHU C W. The Runge-Kutta discontinuous Galerkin method for conservation laws V:Multidimensional systems[J]. J Comput Phys, 1998, 141(2):199-224.
[15] COCKBURN B, SHU C W. Runge-Kutta discontinuous Galerkin method for convection-dominated problems[J]. J Sci Comput, 2001, 16(3):173-261.
[16] BASSI F, REBAY S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations[J]. J Comput Phys, 1997, 131:267-279.
[17] CCCKBURN B, SHU C W. The local discontinuous Galerkin methods for time dependent convection-diffusion equation[J]. SIAM J Numer Anal, 1998, 35:2440-2463.
[18] YAN J, SHU C W. A local discontinuous Galerkin methods for KdV type equaitons[J]. SIAM J Numer Anal, 2002, 40:769-791.
[19] YAN J, SHU C W. Local discontinuous Galerkin methods for partial differential equations with higher order derivatives[J]. J Sci Comput, 2002, 17:27-47.
[20] LIU H L, YAN J. The direct discontinuous Galerkin (DDG) methods for diffusion problems[J]. SIAM J Numer Anal, 2009, 47:475-698.
[21] YE X. A new discontinuous finite volume method for elliptic problems[J]. SIAM J Numer Anal, 2004, 42:1062-1072.
[22] YE X. A discontinuous finite volume method for the Stokes problems[J]. SIAM J Numer Anal, 2006, 44:183-198.
[23] BI C J, GENG J Q. Discontinuous finite volume element method for parabolic problems[J]. Numer Methods Partial Differential Eq, 2010, 26:367-383.
[24] CHEN D W, YU X J. RKCVDFEM for one-dimensional hyperbolic conservation laws[J]. Chinese J Comput Phys, 2009, 26:501-509.
[25] CHEN D W, YU X J, CHEN Z X. The Runge-Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws[J]. Int J Numer Meth Fluids, 2011, 67:711-786.
[26] ZHAO G Z, YU X J. The high order control volume discontinuous Petrov-Galerkin finite element method for the hyperbolic conservation laws based on Lax-Wendroff time discretization[J]. Appl Math Comput, 2015, 252:175-188.
[27] ZHAO G Z, YU X J, GUO H M. A discontinuous Petrov-Galerkin method for the two-dimensional compressible gas dynamic equations in Lagrangian coordinate[J]. Chinese J Comput Phys, 2017, 34(3):294-308.
[28] SHU C W. Total-variation-diminishing time discretizations[J]. SIAM J Sci Stat Comput, 1998, 9:1073-1084.
[29] BAHADIR A R, SAGLAM M. A mixed finite difference and boundary element approach to one-dimensional Burgers equation[J]. Appl Math Comput, 2005, 160:663-673.
[30] BAHADIR A R. Numerical solution for one-dimensional Burgers equation using a fully implicit finite difference method[J]. Appl Math, 1999, 8:897-909.
[31] ZHAO G Z, YU X J, WU D. Numerical solution of the Burgers equation by local discontinuous Galerkin method[J]. Appl Math Comput, 2010, 216:3671-3679.
[32] ZHANG R P, YU X J, ZHAO G Z. Modified Burgers equation by the local discontinuous Galerkin method[J]. Chinese Phys B, 2013, 22(3):030210.
[33] BRATSOS A G. A fourth order numerical scheme for solving the modified Burgers equaiton[J]. Comput Math Appl, 2010, 60:1393-1400.

求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法

A Local Discontinuous Petrov-Galerkin Method for Partial Differential Equations with High Order Derivatives

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

编辑推荐

Metrics

作者中心

审稿中心

期刊浏览

期刊介绍

[1]	赵国忠, 蔚喜军, 董自明, 郭虹平, 郭鹏云, 李姝敏. 非线性Schrödinger方程几类孤立子解: 局部间断Petrov-Galerkin方法[J]. 计算物理, 2022, 39(6): 641-650.
[2]	张欣, 赵国忠, 李宏. 局部间断Petrov-Galerkin方法在大气污染模型中的应用[J]. 计算物理, 2021, 38(2): 171-182.