|
|
Burgers方程的一类三次有限体积元方法
|
Abstract:
本文对Burgers方程的初边值问题，用最佳应力点构建对偶网格剖分，并基于分片三次Lagrange插值试探函数空间和分片常数检验函数空间，构造了Crank-Nicolson三次有限体积元格式并证明了数值解的L2-模最优阶误差估计及其导数在最佳应力节点处的超收敛误差估计。最后，给出数值算例验证了理论分析结果以及所提格式的有效性。
In this paper, for the initial boundary value problem of the Burgers equation, the optimal stress point is used to construct a dual partition, and based on the trial function space of piecewise cubic Lagrange interpolation and the test function space of piecewise constant, the Crank-Nicolson cubic finite volume element scheme is constructed. And the L2 norm optimal order error estimate of the numerical solutions and the super-convergence error estimate of the derivative at the optimal stress node are proved. Finally, numerical examples are given to verify the theoretical analysis re-sults and the validity of the proposed scheme.
| [1] | Hopf, E. (1950) The Partial Difference Equation . Communications on Pure and Applied Mathematics, 3, 201-230. https://doi.org/10.1002/cpa.3160030302 |
| [2] | Cole, J.D. (1951) On a Quasilinear Parabolic Equations Occurring in Aerodynamics. Quarterly of Applied Mathematics, 9, 225-236. https://doi.org/10.1090/qam/42889 |
| [3] | Benton, E.R. and Platzman, G.W. (1972) A Table of Solutions of the One-Dimensional Burgers’ Equations. Quarterly of Applied Mathematics, 30, 195-212. https://doi.org/10.1090/qam/306736 |
| [4] | Chen, C.K., Zhang, X.H. and Liu, Z. (2020) A High-Order Compact Fi-nite Difference Scheme and Precise Integration Method Based on Modified Hopf-Cole Transformation for Numerical Simulation of N-Dimensional Burgers’ System. Applied Mathematics and Computation, 372, Article ID: 125009. https://doi.org/10.1016/j.amc.2019.125009 |
| [5] | 杨梅, 赵凤群, 郭冲. 基于Hopf-Cole变换的Burgers方程初边值问题的Sinc-Galerkin法[J]. 计算力学学报, 2019, 36(6): 807-812. |
| [6] | 高巍, 张宝, 李宏, 刘洋. Burgers方程的高阶紧致有限体积解法[J]. 应用数学, 2016, 29(2): 331-339. |
| [7] | Sheng, Y. and Zhang, T. (2018) The Finite Volume Method for Two-Dimensional Burgers’ Equation. Personal and Ubiquitous Computing, 22, 1133-1139. https://doi.org/10.1007/s00779-018-1143-4 |
| [8] | 宋健, 王天军, 霍金键. Burgers方程的时空Legendre谱配置方法[J]. 应用数学进展, 2021, 10(4), 1380-1386.
https://doi.org/10.12677/AAM.2021.104147 |
| [9] | 陈勋, 蒋艳群, 陈琦, 张旭, 胡迎港. 粘性Burgers方程的高阶精度半隐式WC-NS方法[J]. 数值计算与计算机应用, 2022, 43(1): 77-87. |
| [10] | 胡婷, 傅毛里. 一类空间分数阶Burgers方程守恒型差分方法[J]. 应用数学进展, 2022, 11(1): 219-223.
https://doi.org/10.12677/AAM.2022.111027 |
| [11] | Gao, G.H. and Wang, T.K. (2010) Cubic Superconvergent Fi-nite Volume Element Method for One-Dimensional Elliptic and Parabolic Equations. Journal of Computational and Ap-plied Mathematics, 233, 2285-2301.
https://doi.org/10.1016/j.cam.2009.10.013 |
| [12] | Yu, C.H. and Li, Y.H. (2011) Biquadratic Finite Volume Element Methods Based on Optimal Stress Points for Parabolic Problem. Journal of Computational and Applied Mathematics, 236, 1055-1068.
https://doi.org/10.1016/j.cam.2011.07.030 |
