Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries

Shuang Liu; Xinfeng Liu

doi:10.3934/dcdsb.2019176

Article Contents

2020, Volume 25, Issue 1: 141-159. Doi: 10.3934/dcdsb.2019176

This issue Previous Article Stochastic partial differential equation models for spatially dependent predator-prey equations Next Article Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary

Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries

Shuang Liu and
Xinfeng Liu^,

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

^* Corresponding author: Xinfeng Liu
^* Corresponding author: Xinfeng Liu

Received: November 2018

Revised: February 2019

Early access: July 19, 2019

Published online: July 19, 2019

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Abstract

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population. There are several numerical difficulties to efficiently handle such systems. Firstly extremely small time steps are usually demanded due to the stiffness of the system. Secondly it is always difficult to efficiently and accurately handle the moving boundaries. To overcome these difficulties, we first transform the one-dimensional problem with a moving boundary into a system with a fixed computational domain, and then introduce four different temporal schemes: Runge-Kutta, Crank-Nicolson, implicit integration factor (IIF) and Krylov IIF for handling such stiff systems. Numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches, and it can be shown that Krylov IIF is superior to other three approaches in terms of stability and efficiency by direct comparison.

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Mathematics Subject Classification: Primary: 65N06, 65N40; Secondary: 92D25.

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Figure 1. Error of U as a function of time step sizes

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Figure 2. Error of H as a function of time step sizes

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Figure 3. Solution $ U $ and $ H $ for the large diffusion system

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Figure 4. Solution $ U $ and $ H $ for the stiff system

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Table 1. Convergence test of Runge-Kutta method

$ N_z\times N_t $	$ L_{\infty} Error $	Order	$ L_2 Error $	Order
Accuracy test of W
26$ \times $5e4	1.85e-04		1.32e-04
51$ \times $1e5	4.62e-05	2.00	3.28e-05	2.01
101$ \times $2e5	1.16e-05	1.99	8.22e-06	2.00
201$ \times $4e5	2.89e-06	2.01	2.04e-06	2.01
401$ \times $8e5	6.39e-07	2.18	4.50e-07	2.18
801$ \times $16e5	Reference
Accuracy test of G
26$ \times $5e4	2.66e-04		6.09e-06
51$ \times $1e5	6.65e-05	2.00	1.52e-06	2.00
101$ \times $2e5	1.68e-05	1.99	3.85e-07	1.98
201$ \times $4e5	4.20e-06	2.00	9.65e-08	2.00
401$ \times $8e5	9.38e-07	2.16	2.17e-08	2.15
801$ \times $16e5	Reference

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Table 2. Convergence test of Crank-Nicolson method

$ N_z\times N_t $	$ L_{\infty} Error $	Order	$ L_2 Error $	Order
Accuracy test of W
26$ \times $5e4	1.85e-04		2.68e-04
51$ \times $1e5	4.65e-05	2.00	3.30e-05	1.99
101$ \times $2e5	1.18e-05	1.98	8.30e-06	1.99
201$ \times $4e5	2.92e-06	2.01	2.06e-06	2.01
401$ \times $8e5	6.30e-07	2.21	4.38e-07	2.23
801$ \times $16e5	Reference
Accuracy test of G
26$ \times $5e4	1.33e-04		6.13e-05
51$ \times $1e5	6.72e-05	2.01	1.54e-05	1.99
101$ \times $2e5	1.71e-05	1.98	3.92e-06	1.97
201$ \times $4e5	4.30e-06	1.99	9.90e-07	1.99
401$ \times $8e5	9.30e-07	2.21	2.15e-07	2.20
801$ \times $16e5	Reference

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Table 3. Convergence test of IIF2 method

$ N_z\times N_t $	$ L_{\infty} Error $	Order	$ L_2 Error $	Order
Accuracy test of W
26$ \times $5e4	1.82e-04		1.31e-04
51$ \times $1e5	4.51e-05	2.02	3.20e-05	2.03
101$ \times $2e5	1.11e-05	2.02	7.84e-06	2.03
201$ \times $4e5	2.65e-06	2.07	1.86e-06	2.07
401$ \times $8e5	5.34e-07	2.31	3.73e-07	2.32
801$ \times $16e5		Reference
Accuracy test of G
26$ \times $5e4	2.65e-04		6.07e-06
51$ \times $1e5	6.58e-05	2.01	1.51e-06	2.01
101$ \times $2e5	1.64e-05	2.00	3.78e-07	2.00
201$ \times $4e5	4.00e-06	2.04	9.26e-08	2.03
401$ \times $8e5	8.35e-07	2.26	1.94e-08	2.26
801$ \times $16e5	Reference

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Table 4. Convergence test of Krylov IIF2 method

$ N_z\times N_t $	$ L_{\infty} Error $	Order	$ L_2 Error $	Order
Accuracy test of W
26$ \times $5e4	1.82e-04		1.31e-04
51$ \times $1e5	4.51e-05	2.02	3.20e-05	2.03
101$ \times $2e5	1.11e-05	2.02	7.84e-06	2.03
201$ \times $4e5	2.65e-06	2.07	1.86e-06	2.07
401$ \times $8e5	5.30e-07	2.32	3.72e-07	2.32
801$ \times $16e5	Reference
Accuracy test of G
26$ \times $5e4	2.65e-04		6.07e-06
51$ \times $1e5	6.58e-05	2.01	1.51e-06	2.01
101$ \times $2e5	1.64e-05	2.00	3.78e-07	2.00
201$ \times $4e5	4.00e-06	2.04	9.26e-08	2.03
401$ \times $8e5	8.40e-07	2.25	1.94e-08	2.26
801$ \times $16e5	Reference

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Table 5. Errors and order of accuracy in time for three stable schemes: Crank-Nicolson, IIF2 and Krylov IIF2

$ \triangle t $	Crank-Nicolson		IIF2		Krylov IIF2
Accuracy test of W
	$ L_{\infty} $ error	Order	$ L_{\infty} $ error	Order	$ L_{\infty} $ error	Order
$ 8.0\times10^{-5} $	1.54e-8	-	6.50e-8	-	6.50e-8	-
$ 4.0\times10^{-5} $	4.09e-9	1.91	2.23e-8	1.55	2.23e-8	1.55
$ 2.0\times10^{-5} $	1.05e-9	1.97	6.28e-9	1.83	6.28e-9	1.83
$ 1.0\times10^{-5} $	3.22e-10	1.70	1.30e-9	2.27	1.30e-9	2.27
$ 5.0\times10^{-6} $	8.14e-11	1.98	2.85e-10	2.20	2.85e-10	2.20
$ 2.5\times10^{-6} $	Reference		Reference		Reference
Accuracy test of G
	$ L_{\infty} $ error	Order	$ L_{\infty} $ error	Order	$ L_{\infty} $ error	Order
$ 8.0\times10^{-5} $	4.16e-7	-	8.49e-7	-	8.49e-7	-
$ 4.0\times10^{-5} $	1.44e-7	1.53	2.81e-7	1.59	2.81e-7	1.59
$ 2.0\times10^{-5} $	4.86e-8	1.57	9.22e-8	1.61	9.22e-8	1.61
$ 1.0\times10^{-5} $	1.54e-8	1.66	2.88e-8	1.68	2.88e-8	1.68
$ 5.0\times10^{-6} $	3.92e-9	1.97	7.27e-9	1.99	7.27e-9	1.99
$ 2.5\times10^{-6} $	Reference		Reference		Reference

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Table 6. Efficiency test for the large diffusion system

$ \triangle t=10^{-4} $	$ N_z=1001 $	$ N_z=2001 $	$ N_z=4001 $
Crank-Nicolson	16.576	75.092	395.512
Krylov IIF2	14.595	59.566	295.958
IIF2	211.227	1099.695	9694.277

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Table 7. Efficiency test for the stiff system

$ \triangle t=10^{-4} $	$ N_z=1001 $	$ N_z=2001 $	$ N_z=4001 $
Crank-Nicolson	30.149	141.601	760.832
Krylov IIF2	16.706	71.501	346.278
IIF2	389.514	1877.676	22626.109

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