Bakhvalov-Shishkin网格上求解边界层问题的差分进化算法

摘要/Abstract

摘要：

该文在Bakhvalov-Shishkin网格上求解具有左边界层或右边界层的对流扩散方程，并采用差分进化算法对Bakhvalov-Shishkin网格中的参数进行优化，获得了该网格上具有最优精度的数值解.对三个算例进行了数值模拟，数值结果表明：采用差分进化算法求解具有较高的计算精度和收敛性，特别是边界层的数值解精度明显优于选择固定网格参数时的结果.

关键词: 边界层, Bakhvalov-Shishkin网格, 差分进化算法, 网格参数

Abstract:

In this paper, the convection-diffusion equation with left boundary layer or right boundary layer is solved on Bakhvalov-Shishkin mesh. The parameter in Bakhvalov-Shishkin mesh is optimized by differential evolution algorithm, and we obtain numerical solution with optimal accuracy on Bakhvalov-Shishkin mesh. Three numerical examples are simulated, and the numerical results show that the differential evolution algorithm is accurate and convergence. Especially, the numerical solution accuracy of the boundary layer is obviously better than that of fixed mesh parameters.

Key words: Boundary layer, Bakhvalov-Shishkin mesh, Differential evolution algorithm, Mesh parameter

中图分类号:

O241

周琴,程立正. Bakhvalov-Shishkin网格上求解边界层问题的差分进化算法[J]. 数学物理学报, 2021, 41(1): 245-253.

Qin Zhou,Lizheng Cheng. Differential Evolution Algorithms for Boundary Layer Problems on Bakhvalov-Shishkin Mesh[J]. Acta mathematica scientia,Series A, 2021, 41(1): 245-253.

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ε	N = 32		N = 64		N = 128		N = 256
	误差	β₁	误差	β₁	误差	β₁	误差	β₁
10^-2	3.337E-02	0.269	1.766E-02	0.278	9.068E-03	0.296	4.589E-03	0.293
10^-3	3.578E-02	0.213	1.861E-02	0.255	9.468E-03	0.279	4.774E-03	0.282
10^-4	3.664E-02	0.211	1.896E-02	0.245	9.577E-03	0.265	4.808E-03	0.275
10^-5	3.839E-02	0.173	1.962E-02	0.212	9.778E-03	0.238	4.857E-03	0.261
10^-6	4.023E-02	0.156	2.062E-02	0.183	1.017E-02	0.211	4.986E-03	0.234

ε	N = 32		N = 64		N = 128		N = 256
	误差	β₁	误差	β₁	误差	β₁	误差	β₁
10^-2	2.646E-02	0.842	1.403E-02	0.797	7.218E-03	0.837	3.657E-03	0.826
10^-3	2.790E-02	0.711	1.463E-02	0.729	7.447E-03	0.769	3.749E-03	0.806
10^-4	2.808E-02	0.710	1.471E-02	0.728	7.483E-03	0.768	3.765E-03	0.790
10^-5	2.810E-02	0.709	1.472E-02	0.728	7.487E-03	0.768	3.766E-03	0.790
10^-6	2.810E-02	0.709	1.472E-02	0.728	7.487E-03	0.768	3.767E-03	0.790

ε	N = 32		N = 64		N = 128		N = 256
	误差	β₂	误差	β₂	误差	β₂	误差	β₂
10^-2	5.773E-03	0.441	2.879E-03	0.446	1.393E-03	0.430	6.534E-04	0.397
10^-3	6.997E-03	0.345	3.638E-03	0.357	1.851E-03	0.364	9.320E-04	0.368
10^-4	7.154E-03	0.330	3.733E-03	0.340	1.907E-03	0.346	9.636E-04	0.350
10^-5	7.172E-03	0.328	3.743E-03	0.338	1.913E-03	0.344	9.670E-04	0.347
10^-6	7.173E-03	0.328	3.744E-03	0.338	1.914E-03	0.344	9.673E-04	0.347

网格参数选取	ε = 10^-2		ε = 10^-5
	N = 64	N = 128	N = 64	N = 128
差分进化算法	1.766E-02	9.068E-03	1.962E-02	9.778E-03
β₁ = 2/3	2.486E-02	1.372E-02	3.956E+00	1.012E+00
β₁ = 0.3	1.769E-02	9.075E-03	4.028E-02	1.185E-02
β₁ = 0.1	2.676E-02	1.453E-02	2.734E-02	1.481E-02

网格参数选取	ε = 10^-2		ε = 10^-5
	N = 64	N = 128	N = 64	N = 128
差分进化算法	1.403E-02	7.218E-03	1.472E-02	7.487E-03
β₁ = 1	1.441E-02	7.411E-03	1.580E-02	7.983E-03
β₁ = 0.3	2.085E-02	1.116E-02	2.123E-02	1.135E-02
β₁ = 0.1	4.362E-02	2.522E-02	4.441E-02	2.565E-02