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《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (10): 30-39.doi: 10.6040/j.issn.1671-9352.0.2023.112

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求解广义Burgers-Fisher方程的微分求积法

阿迪力·艾力(),开依沙尔·热合曼*()   

  1. 新疆大学数学与系统科学学院, 新疆 乌鲁木齐 830046
  • 收稿日期:2023-03-13 出版日期:2024-10-20 发布日期:2024-10-10
  • 通讯作者: 开依沙尔·热合曼 E-mail:adili0515@163.com;kaysar106@xju.edu.cn
  • 作者简介:阿迪力·艾力(1998—), 男, 硕士研究生, 研究方向为微分方程数值计算. E-mail: adili0515@163.com
  • 基金资助:
    新疆大学博士启动基金项目(BS150204)

Differential quadrature method for solving the generalized Burgers-Fisher equations

ALI Adil(),RAHMAN Kaysar*()   

  1. College of Mathematics and System Science, Xinjiang University, Urumqi 830046, Xinjiang, China
  • Received:2023-03-13 Online:2024-10-20 Published:2024-10-10
  • Contact: RAHMAN Kaysar E-mail:adili0515@163.com;kaysar106@xju.edu.cn

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摘要：

对Dirichlet边界和Neumann边界条件下的广义Burgers-Fisher方程构造了高精度数值计算格式。首先, 空间上分别采取均匀网格和Chebyshev-Gauss-Lobatto网格的拉格朗日插值多项式微分求积法, 时间上采取三阶强稳定性保持Runge-Kutta格式; 其次, 利用矩阵方法进行稳定性分析; 最后, 对2种不同边界条件下的数值例子进行数值计算, 并将结果和其他数值方法进行比较, 验证本文格式的有效性。

关键词: 广义Burgers-Fisher方程, 微分求积法, Chebyshev-Gauss-Lobatto网格, 强稳定性保持Runge-Kutta格式

Abstract:

In this paper, a high accuracy numerical scheme is constructed for the generalized Burgers-Fisher equation with Dirichlet boundary or Neumann boundary conditions. Firstly, the Lagrange interpolation polynomial differential quadrature method with uniform grid and Chebyshev-Gauss-Lobatto grid is used in space, and the third-order strong stability-preserving Runge-Kutta scheme is used in time. Secondly, the stability of the scheme is analyzed by using the matrix method. Finally, two numerical examples with different boundary conditions are calculated, and the results are compared with other numerical methods to verify the effectiveness of the proposed scheme.

Key words: generalized Burgers-Fisher equation, differential quadrature method, Chebyshev-Gauss-Lobatto grid, strong stability-preserving Runge-Kutta scheme

中图分类号: 

  • O241

图1

当α=1, δ=1, N取不同值时，矩阵A的特征值"

表1

当β=δ=1, N=11, Δt=0.000 1时, 不同x、t和α对应的绝对误差(取CGL网格)"

x t α=-1 α=0.1 α=1
x4 0.3 2.4536e-14 2.2982e-14 2.2649e-14
0.5 2.4647e-14 2.4314e-14 2.3981e-14
0.9 4.1855e-14 2.3759e-14 2.2538e-14
x7 0.3 3.2196e-14 3.1974e-14 3.3418e-14
0.5 3.1641e-14 3.4639e-14 3.6748e-14
0.9 5.4068e-14 3.3307e-14 3.4306e-14
x10 0.3 3.3307e-15 3.5527e-15 4.0523e-15
0.5 3.5527e-15 3.4417e-15 3.9968e-15
0.9 5.3291e-15 3.9968e-15 3.4417e-15

表2

当α=0.1, δ=1, N=11, Δt=0.000 1时, 不同x、t和β对应的绝对误差(取CGL网格)"

x t β=-0.25 β=0.1 β=0.5
x4 0.3 2.1538e-14 2.2649e-14 2.4425e-14
0.5 2.2427e-14 2.3870e-14 2.4758e-14
0.9 2.2260e-14 2.5313e-14 2.4536e-14
x7 0.3 3.0587e-14 3.0753e-14 3.3418e-14
0.5 3.1530e-14 3.6304e-14 3.5305e-14
0.9 3.1142e-14 3.5638e-14 3.5749e-14
x10 0.3 3.2196e-15 3.2196e-15 3.2196e-15
0.5 3.3862e-15 4.1078e-15 3.7748e-15
0.9 3.3862e-15 3.8858e-15 3.6637e-15

表3

当α=1, δ=1, N=11, Δt=0.000 1时, 不同x、t和δ对应的绝对误差(取CGL网格)"

x t δ=2 δ=4 δ=8
x4 0.3 3.0531e-14 2.1871e-14 6.3061e-14
0.5 3.5638e-14 1.5099e-14 8.9928e-15
0.9 3.3640e-14 2.3981e-14 2.2315e-14
x7 0.3 3.9968e-14 3.3973e-14 1.0414e-13
0.5 5.5955e-14 2.5091e-14 2.1760e-14
0.9 5.2514e-14 3.7303e-14 3.3307e-14
x10 0.3 3.6637e-15 5.5511e-16 1.3212e-14
0.5 4.4409e-15 4.7740e-15 2.9976e-15
0.9 6.8834e-15 4.4409e-15 4.3299e-15

表4

当α=1, B=0, δ=0, N=11, Δt=0.000 1时, 不同x、t对应的绝对误差(取均匀网格)"

x t VIM[12] ADM[6] CFD6[3] 本文格式
0.1 1 1.92e-14 1.92e-14 5.77e-15 1.17e-15
0.5 9.73e-14 9.73e-14 4.19e-14 6.10e-14
0.9 1.75e-13 1.75e-13 1.29e-14 4.89e-15
0.1 10 1.63e-12 1.63e-12 8.49e-15 1.16e-14
0.5 9.44e-12 9.44e-12 4.26e-14 4.92e-14
0.9 1.73e-11 1.73e-11 1.82e-14 1.37e-14
0.1 50 8.14e-12 8.14e-12 2.11e-15 2.22e-16
0.5 2.03e-10 2.03e-10 2.46e-14 1.24e-14
0.9 3.99e-10 3.99e-10 8.05e-15 2.22e-16

表5

当α=1, δ=1, N=11, Δt=0.000 1时，两种网格下L∞误差对比"

tΔt=0.000 5 Δt=0.000 1 Δt=0.000 05
均匀网格 CGL网格 均匀网格 CGL网格 均匀网格 CGL网格
0.2 2.7311e-14 2.8311e-14   7.6272e-14 3.3751e-14   1.0614e-13 6.7613e-14
0.5 2.9887e-13 2.8277e-13 1.1069e-13 5.8842e-14 1.4544e-13 1.2157e-13
1.0 4.8095e-13 4.8606e-13 6.7057e-14 5.4068e-14 1.2712e-13 1.2046e-13
1.5 2.7700e-13 2.7789e-13 5.8842e-14 5.6621e-14 1.2257e-13 1.1458e-13

图2

当α=1, β=1, δ=4, N=11, Δt=0.000 1时, 在Dirichlet边界条件下本文数值解和解析解对比(取CGL网格)"

表6

当β=δ=1, N=11, Δt=0.000 1时, 不同x、t和α对应的绝对误差(取CGL网格)"

x t α=-1 α=0.1 α=1
x4 0.3 1.0358e-13 1.0458e-13 1.0725e-13
0.5 1.6331e-13 1.6676e-13 1.7586e-13
0.9 3.1242e-13 2.7667e-13 3.0032e-13
x7 0.3 9.5146e-14 9.7256e-14 9.4702e-14
0.5 1.5454e-13 1.5754e-13 1.6664e-13
0.9 3.0898e-13 2.6912e-13 2.9321e-13
x10 0.3 9.1926e-14 9.5035e-14 9.0372e-14
0.5 1.5177e-13 1.5632e-13 1.6365e-13
0.9 3.0875e-13 2.6779e-13 2.9288e-13

表7

当α=0.1, δ=1, N=11, Δt=0.000 1时, 不同x、t和β对应的绝对误差(取CGL网格)"

x t β=-0.25 β=0.1 β=0.5
x4 0.3 9.6922e-14 9.8810e-14 1.0580e-13
0.5 1.5826e-13 1.6720e-13 1.7264e-13
0.9 2.7855e-13 3.0265e-13 3.0109e-13
x7 0.3 9.1094e-14 9.2260e-14 9.8588e-14
0.5 1.5193e-13 1.6120e-13 1.6631e-13
0.9 2.7156e-13 2.9665e-13 2.9332e-13
x10 0.3 8.9373e-14 8.9539e-14 9.7589e-14
0.5 1.5005e-13 1.5943e-13 1.6276e-13
0.9 2.6951e-13 2.9532e-13 2.9121e-13

表8

当α=1, β=1, N=11, Δt=0.000 1时, 不同x、t和δ对应的绝对误差(取CGL网格)"

x t δ=2 δ=4 δ=8
x4 0.3 1.3445e-13 8.4599e-14 1.5543e-13
0.5 2.0317e-13 1.2434e-13 5.6288e-14
0.9 3.2008e-13 1.3511e-13 7.0166e-14
x7 0.3 1.0625e-13 1.0825e-13 1.8574e-13
0.5 2.0517e-13 1.0592e-13 8.1490e-14
0.9 3.0731e-13 1.2257e-13 6.0618e-14
x10 0.3 9.6145e-14 1.2945e-13 1.9840e-13
0.5 2.1338e-13 9.7811e-14 9.1926e-14
0.9 3.0398e-13 1.1935e-13 5.7288e-14

图3

当α=1, β=1, δ=8, N=11, Δt=0.000 1时，在Neumann边界条件下本文数值解和解析解对比(取CGL网格)"

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