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## Numerical treatment of Gray-Scott model with operator splitting method

 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)

Received  March 2019 Revised  April 2019 Published  September 2020

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.

Citation: Berat Karaagac. Numerical treatment of Gray-Scott model with operator splitting method. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020143
##### References:
 [1] E. N. Aksan, H. Karabenli and A. Esen, An Application Of Finite Element Method For a Moving Boundary Problem, Thermal Science, 22 (2018), 25-32.  doi: 10.2298/TSCI170613268A.  Google Scholar [2] A. H. A. Ali, G. A. Gardner and L. R. T. Gardner, A collocation solution for Burgers' equation using cubic B-spline finite elements, Comput. Methods Appl. Mech. Engrg., 100 (1992), 325-337.  doi: 10.1016/0045-7825(92)90088-2.  Google Scholar [3] İ. Çelikkaya, Operator splitting solution of equal width wave equation based on the Lie-Trotter and strang splitting method, Konuralp J. Math., 6 (2018), 200-208.   Google Scholar [4] X. Cheng, J. Duan and D. Li, A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations, Appl. Math. Comput., 346 (2019), 452-464.  doi: 10.1016/j.amc.2018.10.065.  Google Scholar [5] M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Simulation, 79 (2008), 700-715.  doi: 10.1016/j.matcom.2008.04.018.  Google Scholar [6] F. Dkhil, E. Logak and Y. Nishiura, Some analytical results on the Gray–Scott model, Asymptot. Anal., 39 (2004), 225-261.   Google Scholar [7] A. J. Doelman, T. J. Kaper and P. Zegeling, Pattern formation in the one-dimensional Gray Scott model, Nonlinearity, 10 (1997), 523-563.  doi: 10.1088/0951-7715/10/2/013.  Google Scholar [8] A. J. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [9] A. Esen, O. Tasbozan, Y. Ucar and N. M. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Math. J., 8 (2015), 181-193.  doi: 10.1515/tmj-2015-0020.  Google Scholar [10] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $A + 2B \rightarrow 3B$, $B \rightarrow C$, Chem. Eng. Sci., 39 (1984), 1087-1097.   Google Scholar [11] S. Guo, L. Mei, Z. Zhang, J. Chen, Y. He and Y. Li, Finite difference/Hermite-Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction-diffusion equation in unbounded domains, Appl. Math. Model., 70 (2019), 246-263.  doi: 10.1016/j.apm.2019.01.018.  Google Scholar [12] S. Hasnain, M. Saqib, M. F. Afzaal and N. A. Harbi, Numerical study to coupled three dimensional reaction diffusion system, IEEE Access, 7 (2019), 46695-46705.  doi: 10.1109/ACCESS.2019.2903977.  Google Scholar [13] R. S. Johnson, A Modern Introduction to The Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.  Google Scholar [14] B. Karaagac and A. Esen, The Hunter-Saxton: A numerical approach using collocation method, Numer. Methods Partial Differential Equations, 34 (2018), 1637-1644.  doi: 10.1002/num.22199.  Google Scholar [15] A. H. Khater, R. S. Temsah and M. M. Hassan, A Chebyshev spectral collocation method for solving Burgers-type equations, J. Comput. Appl. Math., 222 (2008), 333-350.  doi: 10.1016/j.cam.2007.11.007.  Google Scholar [16] Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, an Diego, 2003.   Google Scholar [17] V. V. Konotop, Dark solitons in Bose-Einstein condensates: Theory, Springer, (2008), 65–83. Google Scholar [18] A. Korkmaz, O. Ersoy Hepson and ł. Dag, Motion of patterns modeled by the Gray-Scott autocatalysis system in one dimension, MATCH Commun. Math. Comput. Chem., 77 (2017), 507-526.   Google Scholar [19] S. Kumar, R. Jiwari and R. C. Mittal, Numerical simulation for computational modelling of reaction-diffusion Brusselator model arising in chemical processes, J. Math. Chem., 57 (2019), 149-179.  doi: 10.1007/s10910-018-0941-2.  Google Scholar [20] S. Kutluay, Y. Ucar and N. M. Yagmurlu, Numerical solutions of the modified Burgers equation by a cubic B-spline collocation method, Bull. Malays. Math. Sci. Soc., 39 (2016), 1603-1614.  doi: 10.1007/s40840-015-0262-6.  Google Scholar [21] S. A. Manaa and J. Rasheed, Successive and finite difference method for gray Scott model, Science Journal of University of Zakho, 1 (2013), 862-873.   Google Scholar [22] G. Micula and S. Micula, Handbook of Splines, Mathematics and its Applications, 462, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-011-5338-6.  Google Scholar [23] R. C. Mittal and R. K. Jain, Numerical solutions of nonlinear Burgers equation with modified cubic B-splines collocation method, Appl. Math. Comput., 218 (2012), 7839-7855.  doi: 10.1016/j.amc.2012.01.059.  Google Scholar [24] A. K. Mittal and V. K. Kukreja, Solution of Burger's equations by orthogonal collocation on finite elements hermite basis, 6th Int. Conference On Advances in Engineering Sciences and Applied Mathematics (Icaesam'2016) Dec. Kuala Lumpur (Malaysia), 2016, 40–45. Google Scholar [25] A. T. Onarcan, N. Adar and I. Dag, Numerical solutions of reaction-diffusion equation systems with trigonometric quintic B-spline collocation algorithm, preprint, arXiv: 1701.04558. Google Scholar [26] A. B. Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT, 54 (2014), 937-954.  doi: 10.1007/s10543-014-0484-2.  Google Scholar [27] O. Oruc, A. Esen and F. Bulut, A Haar wavelet collocation method for coupled nonlinear Schrödinger- KdV equations, Internat. J. Modern Phys., 27 (2016), 1650103, 16 pp. doi: 10.1142/S0129183116501035.  Google Scholar [28] K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, SpringerPlus, 5 (2016), 2-19.  doi: 10.1186/s40064-016-3295-x.  Google Scholar [29] K. M. Owolabi and K. C. Patidar, Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology, Appl. Math. Comput., 240 (2014), 30-50.  doi: 10.1016/j.amc.2014.04.055.  Google Scholar [30] K. M. Owolabi, Numerical analysis and pattern formation process for space-fractional superdiffusive systems, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 543-566.   Google Scholar [31] K. M. Owolabi, Robust IMEX schemes for solving two-dimensional reaction-diffusion models, Int. J. Nonlinear Sci. Numer. Simul., 16 (2015), 271-284.  doi: 10.1515/ijnsns-2015-0004.  Google Scholar [32] K. M. Owolabi and A. Atangana, Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction-diffusion systems, Comput. Appl. Math., 37 (2018), 2166-2189.  doi: 10.1007/s40314-017-0445-x.  Google Scholar [33] K. M. Owolabi and E. Pindza, Mathematical and computational studies of fractional reaction-diffusion system modelling predator-prey interactions, J. Numer. Math., 26 (2018), 97-110.   Google Scholar [34] K. M. Owolabi and K. C. Patidar, Numerical solution of singular patterns in one-dimensional Gray-Scott-like models, Int. J. Nonlinear Sci. Numer. Simul., 15 (2014), 437-462.  doi: 10.1515/ijnsns-2013-0124.  Google Scholar [35] J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192.  doi: 10.1126/science.261.5118.189.  Google Scholar [36] L. A. Peletier, Pulses, kinks and fronts in the Gray-Scott model, Nonlinear Diffusive Systems-dynamics and Asymptotic Analysis (Japanese) (Kyoto, 2000), Surikaisekikenkyusho Kokyuroku, 1178 (2000), 16–28.  Google Scholar [37] S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, On the solutions of time-fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3847-3854.  doi: 10.1016/j.cnsns.2010.02.007.  Google Scholar [38] M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696.  doi: 10.1007/BF03167410.  Google Scholar [39] S. G. Rubin and R. A. Graves, A Cubic Spline Approximation for Problems in Fluid Mechanics, NASA TR R-436, Washington, DC, 1975. Google Scholar [40] N. Stollenwerk and J. P. Boto, Reaction-superdiffusion systems in epidemiology, an application of fractional calculus, AIP Conf. Proc., 1168 (2009), 1548-1551.  doi: 10.1063/1.3241397.  Google Scholar [41] V. Tuoi, Mathematical Analysis of Some Models for Drug Delivery, Phd thesis, National University of Ireland, 2012. Google Scholar [42] A. M. Turing, The chemical basis of morphogenesis, Bulletin of Mathematical Biology, 52 (1990), 153-197.   Google Scholar [43] Y. Ucar, N. M. Yagmurlu and İ. Çelikkaya, Operator splitting for numerical solution of the modified Burgers' equation using finite element method, Numer. Methods Partial Differential Equations, 35 (2019), 478-492.  doi: 10.1002/num.22309.  Google Scholar [44] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004.  Google Scholar [45] A. M. Wazwaz, New solitary wave solutions to the modified Kawahara equation, Phys. Lett. A, 360 (2007), 588-592.  doi: 10.1016/j.physleta.2006.08.068.  Google Scholar [46] O. P. Yadav and R. Jiwari, A finite element approach for analysis and computational modelling of coupled reaction diffusion models, Numer. Methods Partial Differential Equations, 35 (2019), 830-850.  doi: 10.1002/num.22328.  Google Scholar

show all references

##### References:
 [1] E. N. Aksan, H. Karabenli and A. Esen, An Application Of Finite Element Method For a Moving Boundary Problem, Thermal Science, 22 (2018), 25-32.  doi: 10.2298/TSCI170613268A.  Google Scholar [2] A. H. A. Ali, G. A. Gardner and L. R. T. Gardner, A collocation solution for Burgers' equation using cubic B-spline finite elements, Comput. Methods Appl. Mech. Engrg., 100 (1992), 325-337.  doi: 10.1016/0045-7825(92)90088-2.  Google Scholar [3] İ. Çelikkaya, Operator splitting solution of equal width wave equation based on the Lie-Trotter and strang splitting method, Konuralp J. Math., 6 (2018), 200-208.   Google Scholar [4] X. Cheng, J. Duan and D. Li, A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations, Appl. Math. Comput., 346 (2019), 452-464.  doi: 10.1016/j.amc.2018.10.065.  Google Scholar [5] M. Dehghan and A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. Comput. Simulation, 79 (2008), 700-715.  doi: 10.1016/j.matcom.2008.04.018.  Google Scholar [6] F. Dkhil, E. Logak and Y. Nishiura, Some analytical results on the Gray–Scott model, Asymptot. Anal., 39 (2004), 225-261.   Google Scholar [7] A. J. Doelman, T. J. Kaper and P. Zegeling, Pattern formation in the one-dimensional Gray Scott model, Nonlinearity, 10 (1997), 523-563.  doi: 10.1088/0951-7715/10/2/013.  Google Scholar [8] A. J. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotics approach, Phys. D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [9] A. Esen, O. Tasbozan, Y. Ucar and N. M. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Math. J., 8 (2015), 181-193.  doi: 10.1515/tmj-2015-0020.  Google Scholar [10] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system $A + 2B \rightarrow 3B$, $B \rightarrow C$, Chem. Eng. Sci., 39 (1984), 1087-1097.   Google Scholar [11] S. Guo, L. Mei, Z. Zhang, J. Chen, Y. He and Y. Li, Finite difference/Hermite-Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction-diffusion equation in unbounded domains, Appl. Math. Model., 70 (2019), 246-263.  doi: 10.1016/j.apm.2019.01.018.  Google Scholar [12] S. Hasnain, M. Saqib, M. F. Afzaal and N. A. Harbi, Numerical study to coupled three dimensional reaction diffusion system, IEEE Access, 7 (2019), 46695-46705.  doi: 10.1109/ACCESS.2019.2903977.  Google Scholar [13] R. S. Johnson, A Modern Introduction to The Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511624056.  Google Scholar [14] B. Karaagac and A. Esen, The Hunter-Saxton: A numerical approach using collocation method, Numer. Methods Partial Differential Equations, 34 (2018), 1637-1644.  doi: 10.1002/num.22199.  Google Scholar [15] A. H. Khater, R. S. Temsah and M. M. Hassan, A Chebyshev spectral collocation method for solving Burgers-type equations, J. Comput. Appl. Math., 222 (2008), 333-350.  doi: 10.1016/j.cam.2007.11.007.  Google Scholar [16] Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, an Diego, 2003.   Google Scholar [17] V. V. Konotop, Dark solitons in Bose-Einstein condensates: Theory, Springer, (2008), 65–83. Google Scholar [18] A. Korkmaz, O. Ersoy Hepson and ł. Dag, Motion of patterns modeled by the Gray-Scott autocatalysis system in one dimension, MATCH Commun. Math. Comput. Chem., 77 (2017), 507-526.   Google Scholar [19] S. Kumar, R. Jiwari and R. C. Mittal, Numerical simulation for computational modelling of reaction-diffusion Brusselator model arising in chemical processes, J. Math. Chem., 57 (2019), 149-179.  doi: 10.1007/s10910-018-0941-2.  Google Scholar [20] S. Kutluay, Y. Ucar and N. M. Yagmurlu, Numerical solutions of the modified Burgers equation by a cubic B-spline collocation method, Bull. Malays. Math. Sci. Soc., 39 (2016), 1603-1614.  doi: 10.1007/s40840-015-0262-6.  Google Scholar [21] S. A. Manaa and J. Rasheed, Successive and finite difference method for gray Scott model, Science Journal of University of Zakho, 1 (2013), 862-873.   Google Scholar [22] G. Micula and S. Micula, Handbook of Splines, Mathematics and its Applications, 462, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-011-5338-6.  Google Scholar [23] R. C. Mittal and R. K. Jain, Numerical solutions of nonlinear Burgers equation with modified cubic B-splines collocation method, Appl. Math. Comput., 218 (2012), 7839-7855.  doi: 10.1016/j.amc.2012.01.059.  Google Scholar [24] A. K. Mittal and V. K. Kukreja, Solution of Burger's equations by orthogonal collocation on finite elements hermite basis, 6th Int. Conference On Advances in Engineering Sciences and Applied Mathematics (Icaesam'2016) Dec. Kuala Lumpur (Malaysia), 2016, 40–45. Google Scholar [25] A. T. Onarcan, N. Adar and I. Dag, Numerical solutions of reaction-diffusion equation systems with trigonometric quintic B-spline collocation algorithm, preprint, arXiv: 1701.04558. Google Scholar [26] A. B. Orovio, D. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT, 54 (2014), 937-954.  doi: 10.1007/s10543-014-0484-2.  Google Scholar [27] O. Oruc, A. Esen and F. Bulut, A Haar wavelet collocation method for coupled nonlinear Schrödinger- KdV equations, Internat. J. Modern Phys., 27 (2016), 1650103, 16 pp. doi: 10.1142/S0129183116501035.  Google Scholar [28] K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, SpringerPlus, 5 (2016), 2-19.  doi: 10.1186/s40064-016-3295-x.  Google Scholar [29] K. M. Owolabi and K. C. Patidar, Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology, Appl. Math. Comput., 240 (2014), 30-50.  doi: 10.1016/j.amc.2014.04.055.  Google Scholar [30] K. M. Owolabi, Numerical analysis and pattern formation process for space-fractional superdiffusive systems, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 543-566.   Google Scholar [31] K. M. Owolabi, Robust IMEX schemes for solving two-dimensional reaction-diffusion models, Int. J. Nonlinear Sci. Numer. Simul., 16 (2015), 271-284.  doi: 10.1515/ijnsns-2015-0004.  Google Scholar [32] K. M. Owolabi and A. Atangana, Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction-diffusion systems, Comput. Appl. Math., 37 (2018), 2166-2189.  doi: 10.1007/s40314-017-0445-x.  Google Scholar [33] K. M. Owolabi and E. Pindza, Mathematical and computational studies of fractional reaction-diffusion system modelling predator-prey interactions, J. Numer. Math., 26 (2018), 97-110.   Google Scholar [34] K. M. Owolabi and K. C. Patidar, Numerical solution of singular patterns in one-dimensional Gray-Scott-like models, Int. J. Nonlinear Sci. Numer. Simul., 15 (2014), 437-462.  doi: 10.1515/ijnsns-2013-0124.  Google Scholar [35] J. E. Pearson, Complex patterns in a simple system, Science, 261 (1993), 189-192.  doi: 10.1126/science.261.5118.189.  Google Scholar [36] L. A. Peletier, Pulses, kinks and fronts in the Gray-Scott model, Nonlinear Diffusive Systems-dynamics and Asymptotic Analysis (Japanese) (Kyoto, 2000), Surikaisekikenkyusho Kokyuroku, 1178 (2000), 16–28.  Google Scholar [37] S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, On the solutions of time-fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 3847-3854.  doi: 10.1016/j.cnsns.2010.02.007.  Google Scholar [38] M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657-696.  doi: 10.1007/BF03167410.  Google Scholar [39] S. G. Rubin and R. A. Graves, A Cubic Spline Approximation for Problems in Fluid Mechanics, NASA TR R-436, Washington, DC, 1975. Google Scholar [40] N. Stollenwerk and J. P. Boto, Reaction-superdiffusion systems in epidemiology, an application of fractional calculus, AIP Conf. Proc., 1168 (2009), 1548-1551.  doi: 10.1063/1.3241397.  Google Scholar [41] V. Tuoi, Mathematical Analysis of Some Models for Drug Delivery, Phd thesis, National University of Ireland, 2012. Google Scholar [42] A. M. Turing, The chemical basis of morphogenesis, Bulletin of Mathematical Biology, 52 (1990), 153-197.   Google Scholar [43] Y. Ucar, N. M. Yagmurlu and İ. Çelikkaya, Operator splitting for numerical solution of the modified Burgers' equation using finite element method, Numer. Methods Partial Differential Equations, 35 (2019), 478-492.  doi: 10.1002/num.22309.  Google Scholar [44] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004.  Google Scholar [45] A. M. Wazwaz, New solitary wave solutions to the modified Kawahara equation, Phys. Lett. A, 360 (2007), 588-592.  doi: 10.1016/j.physleta.2006.08.068.  Google Scholar [46] O. P. Yadav and R. Jiwari, A finite element approach for analysis and computational modelling of coupled reaction diffusion models, Numer. Methods Partial Differential Equations, 35 (2019), 830-850.  doi: 10.1002/num.22328.  Google Scholar
Numerical simulation of Gray Scott model
Numerical simulation of Gray Scott model
Numerical simulation of Gray Scott model for $u_{apprx}\left( x, t\right)$ and $v_{apprx}\left( x, t\right)$
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
 $u\left( x, t\right)$ $v\left( x, t\right)$ $h$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ 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
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
 $u\left( x, t\right)$ $v\left( x, t\right)$ $h$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ $L_{2}\times 10^{6}$ $L_{\infty }\times 10^{6}$ 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
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
 $u\left( x, t\right)$ $v\left( x, t\right)$ 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
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
 $u\left( x, t\right)$ $v\left( x, t\right)$ 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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