Numerical treatment of Gray-Scott model with operator splitting method

Berat Karaagac

doi:10.3934/dcdss.2020143

Article Contents

2021, Volume 14, Issue 7: 2373-2386. Doi: 10.3934/dcdss.2020143

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Numerical treatment of Gray-Scott model with operator splitting method

Berat Karaagac^1,2, ,

1.
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2.
Department of Mathematics Education, Adıyaman University, Adıyaman, Turkey

^* Corresponding author: Berat Karaagac(beratkaraagac@tdtu.edu.vn)
^* Corresponding author: Berat Karaagac(beratkaraagac@tdtu.edu.vn)

Received: February 28, 2019

Revised: March 31, 2019

Early access: September 2020

Published: July 2021

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Abstract

This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, "strang splitting" idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms $ L_2 $ and $ L_\infty $. Numerical results arising from the simulation experiments are also presented.

Keywords:

Mathematics Subject Classification: Primary: 65L60, 35Q68; Secondary: 65D07.

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Figure 1. Numerical simulation of Gray Scott model

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Figure 2. Numerical simulation of Gray Scott model

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Figure 3. Numerical simulation of Gray Scott model for $ u_{apprx}\left( x, t\right) $ and $ v_{apprx}\left( x, t\right) $

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Table 1. Gray Scott model: The error norms for $ \Delta t = 0.01 $ and various values of $ h $ at $ T = 1 $

$ u\left( x, t\right) $			$ v\left( x, t\right) $
$ h $	$ L_{2}\times 10^{6} $	$ L_{\infty }\times 10^{6} $	$ L_{2}\times 10^{6} $	$ L_{\infty }\times 10^{6} $
0.1	16.255794672	8.8693520879	16.255794674	8.8693521167
0.05	4.7212245929	3.5788660872	4.7212245653	3.5788660765
0.025	3.1114886810	3.6707532707	3.1114889322	3.6707531289
0.0125	3.3850799985	9.7614962725	3.3850788862	9.7614970661

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Table 2. Gray Scott model: The error norms for $ \Delta t = 0.0025 $ and various values of $ h $ at $ T = 1 $

$ u\left( x, t\right) $			$ v\left( x, t\right) $
$ h $	$ L_{2}\times 10^{6} $	$ L_{\infty }\times 10^{6} $	$ L_{2}\times 10^{6} $	$ L_{\infty }\times 10^{6} $
0.1	16.339991746	8.7969669421	16.339991702	8.7969670275
0.05	4.8404628306	3.5790496331	4.8404627644	3.5790495422
0.025	2.9993515295	3.6711268925	2.9993514761	3.6711268910
0.0125	2.9168925317	3.7175342072	2.9168926962	3.7175350072

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Table 3. Gray Scott model: The error norms for $ \Delta t = 0.00125 $ and various values of $ h $ at $ T = 1 $

$ u\left( x, t\right) $			$ v\left( x, t\right) $
h	$ L_{2}\times 10^{6} $	$ L_{\infty }\times 10^{6} $	$ L_{2}\times10^{6} $	$ L_{\infty }\times 10^{6} $
0.1	16.374721220	8.7933451145	16.374721299	8.7933453672
0.05	4.9049725941	3.5790587130	4.9049726730	3.5790587406
0.025	3.0513500430	3.6711456886	3.0513501767	3.6711458248
0.0125	3.0513500430	3.6711456886	3.0513501767	3.6711458248

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Table 4. Gray Scott model: The error norms for $ \Delta t = 0.001 $ and various values of $ h $ at $ T = 1 $

$ u\left( x, t\right) $			$ v\left( x, t\right) $
h	$ L_{2}\times 10^{6} $	$ L_{\infty }\times 10^{6} $	$ L_{2}\times 10^{6} $	$ L_{\infty }\times 10^{6} $
0.1	16.382270949	8.7929105788	16.382270862	8.7929107112
0.05	4.9186006922	3.5790599957	4.9186005560	3.5790598445
0.025	3.0624459954	3.6711482907	3.0624458638	3.6711480981
0.0125	2.9466605008	3.7175784792	2.9466601800	3.7175779681

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