Differential-spectral approximation based on reduced-dimension scheme for fourth-order parabolic equation

Zhenlan Pan; Jing An

doi:10.3934/dcdsb.2023207

Article Contents

2024, Volume 29, Issue 7: 2928-2946. Doi: 10.3934/dcdsb.2023207

This issue Previous Article On the global existence of mild solutions to binary-ternary Boltzmann equation in an integrable space Next Article The number of limit cycles of Josephson equation

Differential-spectral approximation based on reduced-dimension scheme for fourth-order parabolic equation

Zhenlan Pan and
Jing An^,

School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China

^*Corresponding author: Jing An
^*Corresponding author: Jing An

Received: September 2023

Revised: November 2023

Early access: December 2023

Published: July 2024

This work is supported in part by National Natural Science Foundation of China (No. 12061023), Natural Science Research Project of Guizhou Provincial Department of Education (No. QJJ [2023] 011).

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Abstract

We propose in this paper an efficient differential-spectral approximation based on a reduced-dimension scheme for a fourth-order parabolic equation in a circular domain. First, we decompose the original problem into a series of equivalent one-dimensional fourth-order parabolic problems, based on which a fully discrete scheme based on differential-spectral approximation is established, and its stability and corresponding error estimation are also proved. Then, we utilized the orthogonality of Legendre polynomials to construct a set of effective basis functions and derived the matrix form associated with the full discrete scheme. Finally, several numerical examples are performed, and the numerical results account for the effectiveness and high accuracy of our algorithm.

Keywords:

Mathematics Subject Classification: Primary: 65N15; Secondary: 65N35.

Citation:

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Figure 1. Images of $ u(x, y, t_{1000}) $ (left) and $ u_{MN}^{1000}(x, y) $ (right) at $ t = 1 $ with $ N $ = 30 and $ M $ = 12

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Figure 2. Error between $ u(x, y, t_{1000}) $ and $ u_{MN}^{1000}(x, y) $ with $ N $ = 20 and $ M $ = 10 (left) and $ N $ = 40 and $ M $ = 15 (right) at $ t = 1 $

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Figure 3. Images of $ u(x, y, t_{5000}) $ (left) and $ u_{MN}^{5000}(x, y) $ (right) at $ t = 5 $ with $ N $ = 30 and $ M $ = 15

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Figure 4. Error between $ u(x, y, t_{5000}) $ and $ u_{MN}^{5000}(x, y) $ with $ N $ = 40 and $ M $ = 10 (left) and $ N $ = 30 and $ M $ = 10 (right) at $ t = 5 $

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Figure 5. Error curves on log-log scale under $ L^\infty $ norm for fixed $ N = 20 $ and different $ \tau $ (left) and fixed $ \tau = 0.001 $ and different $ N $ (right)

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Figure 6. Images of $ u(x, y, t_{1000}) $ (left) and $ u_{MN}^{1000}(x, y) $ (right) at $ t = 1 $ with $ N $ = 30 and $ M $ = 10

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Figure 7. Error between $ u(x, y, t_{1000}) $ and $ u_{MN}^{1000}(x, y) $ with $ N $ = 30 and $ M $ = 8 (left) and $ N $ = 40 and $ M $ = 10 (right) at $ t = 1 $

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Figure 8. Images of $ u(x, y, t_{5000}) $ (left) and $ u_{MN}^{5000}(x, y) $ (right) at $ t = 5 $ with $ N $ = 30 and $ M $ = 14

Download: Full-size image PowerPoint slide

Figure 9. Error between $ u(x, y, t_{5000}) $ and $ u_{MN}^{5000}(x, y) $ with $ N $ = 20 and $ M $ = 10 (left) and $ N $ = 40 and $ M $ = 8 (right) at $ t = 5 $

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Table 1. Error at $ t = 1 $ for different $ M $ and $ N $

$ N $	$ M=4 $	$ M=6 $	$ M=8 $	$ M=10 $
20	0.0013	5.7562e-05	1.5984e-05	1.5984e-05
30	0.0013	5.7456e-05	1.5984e-05	1.5984e-05
40	0.0013	5.7886e-05	1.5984e-05	1.5984e-05

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Table 2. Error at $ t = 5 $ for different $ M $ and $ N $

$ N $	$ M=4 $	$ M=6 $	$ M=8 $	$ M=10 $
20	0.0718	0.0031	8.7271e-04	8.7271e-04
30	0.0724	0.0031	8.7272e-04	8.7272e-04
40	0.0723	0.0031	8.7272e-04	8.7272e-04

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Table 3. Error at $ t = 1 $ for different $ M $ and $ N $

$ N $	$ M=4 $	$ M=6 $	$ M=8 $	$ M=10 $
20	9.8927e-04	8.9926e-05	6.4376e-06	6.2753e-06
30	9.9563e-04	9.007e-05	6.4771e-06	3.9904e-06
40	9.9350e-04	8.9994e-05	6.4615e-06	3.9904e-06

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Table 4. Error at $ t = 5 $ for different $ M $ and $ N $

$ N $	$ M=4 $	$ M=6 $	$ M=8 $	$ M=10 $
20	0.0014	3.1297e-05	8.0230e-06	7.9839e-06
30	0.0014	3.1579e-05	7.8656e-06	3.5081e-06
40	0.0014	9.1626e-05	7.9503e-06	3.5081e-06

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