A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations
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摘要:
Burgers方程是一类应用广泛的非线性偏微分方程，方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步：第一步，对三维Burgers方程中的线性导数项(包括时间导数项)，求出相应的多项式特解。第二步，将求出的多项式特解作为基函数，对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比，该算法简单易行，得到的近似解精度非常高，算法极其稳定，对于教学过程中提高学生的编程能力，加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。
Abstract:As a class of nonlinear partial differential equations, the Burgers equations are widely used in various fields. A new space-time polynomial collocation method was presented for particular solutions to 3D Burgers equations. The basic process was divided into 2 steps. The 1st step is to find the polynomial particular solutions of the linear differential operator terms (including the time differential term) in the governing equation. The 2nd step is to solve the nonlinear term of the 3D Burgers equation iteratively. The proposed method is simple and easy to program. The approximate solution has high accuracy. Especially, the stability of the method is excellent, which improves the programming simplicity and deepens the understanding of high-dimensional Burgers equations and the practical application.
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Key words:
- collocation method /
- polynomial solution method /
- Burgers equation /
- MATLAB program
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图 1 配点图：(a) 正方形区域规则取点；(b) 星形区域规则取点；(c) 正方形区域随机取点；(d) 星形区域随机取点(篮圈：内点；红星：边界点)
Figure 1. The collocation point diagram: (a) the regular domain with regular points; (b) the irregular domain with regular points; (c) the regular domain with random points; (d) the irregular domain with random points (blue circles: interior collocation points; red stars: boundary collocation points)
图 2 时空区域中星形区域随机取点（篮圈：内点；红星：边界点）
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同。
Figure 2. An irregular domain with random points in the space-time domain (blue circles: interior collocation points; red stars: boundary collocation points)
图 3 空间求解区域：(a) 正方体区域；(b) bumpy-shaped区域
Figure 3. The spatial solution domains: (a) the cubic domain; (b) the bumpy-shaped domain
图 6
${\tau}=2$ 时，正方体区域上系数矩阵的条件数：(a) 多项式特解系数矩阵的条件数；(b) 多项式基函数系数矩阵的条件数Figure 6. Matrix condition numbers of the cubic domain for
${\tau}=2$ : (a) the condition number of polynomial particular solutions; (b) the condition number of polynomial basis functions表 1
$n_{{{\rm{i}}}}=3\;819,\;n_{{{\rm{d}}}}=5\;220,\;n_{{{\rm{t}}}}=1\;919,\;{\tau}=1$ 时，两种区域多项式特解的平方根误差Table 1. The RMSE with
$n_{{{\rm{i}}}}=3\;819,\; n_{{{\rm{d}}}}=5\;220,\; n_{{{\rm{t}}}}=1\;919,\; {\tau}=1$ for cubic and bumpy-shaped domainsorder cubic ${\delta_{\rm{RMSE1}}}$ bumpy-shape ${\delta_{\rm{RMSE2}}}$ CPU time t/s 4 7.65E−4 1.21E−4 0.375 6 4.74E−5 1.84E−6 1.886 8 2.42E−6 5.13E−9 10.290 10 1.31E−7 5.13E−10 38.707 
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表 2
$n_{{{\rm{i}}}}=3\;819,\;n_{{{\rm{d}}}}=5\;220,\;n_{{{\rm{t}}}}=1\;919,\;{\tau}=1$ 时，两种区域多项式基函数的平方根误差Table 2. The RMSE with
$n_{{{\rm{i}}}}=3\;819,\; n_{{{\rm{d}}}}=5\;220,\; n_{{{\rm{t}}}}=1\;919, \;{\tau}=1$ for cubic and bumpy-shaped domainsorder cubic $ {\delta_{\rm{RMSE1} } }$ bumpy-shape $ {\delta_{\rm{RMSE2} } }$ CPU time t/s 4 2.14E−2 8.22E−2 0.454 6 2.04E−2 7.70E−2 3.418 8 2.02E−2 5.53E−2 23.440 10 2.00E−2 5.51E−2 126.793
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表 3
$n_{{{\rm{i}}}}=3\;819,\;n_{{{\rm{d}}}}=5\;220,\;n_{{{\rm{t}}}}=1\;919,\;\mu=0.5$ 时，两种区域多项式基函数的平方根误差和计算所需时间Table 3. The RMSE with
$n_{{{\rm{i}}}}=3\;819, n_{{{\rm{d}}}}=5\;220, n_{{{\rm{t}}}}=1\;919, \mu=0.5$ for cubic and bumpy-shaped domainsorder cubic $\delta_{{\rm{RMSE}}1}$ bumpy-shape $\delta_{{\rm{RMSE}}2}$ CPU time t/s 4 4.44E−3 3.49E−2 0.590 6 4.40E−3 9.85E−3 2.367 8 4.35E−3 5.78E−3 9.780 10 4.32E−3 7.27E−3 37.786
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