# American Institute of Mathematical Sciences

January  2022, 27(1): 601-617. doi: 10.3934/dcdsb.2021057

## A meshless collocation method with a global refinement strategy for reaction-diffusion systems on evolving domains

 1 Taiyuan University of Technology, Taiyuan, Shanxi Province, China 2 SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong Province, China 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

* Corresponding author

Received  August 2020 Revised  December 2020 Published  January 2022 Early access  February 2021

Turing-type reaction-diffusion systems on evolving domains arising in biology, chemistry and physics are considered in this paper. The evolving domain is transformed into a reference domain, on which we use a second order semi-implicit backward difference formula (SBDF2) for time integration and a meshless collocation method for space discretization. A global refinement strategy is proposed to reduce the computational cost. Numerical experiments are carried out for different evolving domains. The convergence behavior of the proposed algorithm and the effectiveness of the refinement strategy are verified numerically.

Citation: Siqing Li, Zhonghua Qiao. A meshless collocation method with a global refinement strategy for reaction-diffusion systems on evolving domains. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 601-617. doi: 10.3934/dcdsb.2021057
##### References:
 [1] A. Alphonse, C. M. Elliott and B. Stinner, An abstract framework for parabolic PDEs on evolving spaces, Port. Math., 72 (2015), 1–46. doi: 10.4171/PM/1955. [2] S. Bonaccorsi and G. Guatteri, A variational approach to evolution problems with variable domains, Journal of Differential Equations, 175 (2001), 51-70.  doi: 10.1006/jdeq.2000.3959. [3] K. C. Cheung, L. Ling and R. Schaback, $H^2$-Convergence of least-squares kernel collocation methods, SIAM Journal on Numerical Analysis, 56 (2018), 614-633. doi: 10.1137/16M1072863. [4] E. J. Crampin, E. A. Gaffney and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bulletin of Mathematical Biology, 61 (1999), 1093-1120.  doi: 10.1006/bulm.1999.0131. [5] E. J. Crampin, W. W. Hackborn and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bulletin of Mathematical Biology, 64 (2002), 747-769. doi: 10.1006/bulm.2002.0295. [6] M. Dehghan, M. Abbaszadeh and A. Mohebbi, A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model, Engineering Analysis with Boundary Elements, 56 (2015), 129-144.  doi: 10.1016/j.enganabound.2015.02.005. [7] D. Edelmann, Finite element analysis for a diffusion equation on a harmonically evolving domain, preprint, arXiv: 2009.11105. [8] R. I. Fernandes, B. Bialecki and G. Fairweather, An ADI extrapolated Crank–Nicolson orthogonal spline collocation method for nonlinear reaction–diffusion systems on evolving domains, Journal of Computational Physics, 299 (2015), 561-580.  doi: 10.1016/j.jcp.2015.07.016. [9] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234. [10] L. A González, J. C Vanegas and D. A Garzón, Formación de patrones en sistemas de reacción-difusión en dominios crecientes, Revista Internacional de Métodos Numéricos, 25 (2009), 145–161. [11] P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behavior in isothermal reactions, Journal of Physical Chemistry, 89 (1985), 22-32.  doi: 10.1021/j100247a009. [12] Y. C. Hon and R. Schaback, On unsymmetric collocation by radial basis functions, Applied Mathematics and Computation, 119 (2001), 177-186.  doi: 10.1016/S0096-3003(99)00255-6. [13] G. Hu, Z. Qiao and T. Tang, Moving finite element simulations for reaction-diffusion systems, Advances in Applied Mathematics & Mechanics, 4 (2012), 365-381. doi: 10.4208/aamm.10-m11180. [14] E. J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates, Computers & Mathematics with Applications, 19 (1990), 127-145.  doi: 10.1016/0898-1221(90)90270-T. [15] E. J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & Mathematics with Applications, 19 (1990), 147-161.  doi: 10.1016/0898-1221(90)90271-K. [16] P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, Journal of Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017. [17] S. Kondo and R. Asai, A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0. [18] S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.  doi: 10.1126/science.1179047. [19] O. Lakkis, A. Madzvamuse and C. Venkataraman, Implicit–explicit timestepping with finite element approximation of reaction–diffusion systems on evolving domains, SIAM Journal on Numerical Analysis, 51 (2013), 2309-2330.  doi: 10.1137/120880112. [20] W. Li, K. Rubasinghe, G. Yao and L. H. Kuo, The modified localized method of approximated particular solutions for linear and nonlinear convection-diffusion-reaction PDEs, Advances in Applied Mathematics and Mechanics, 12 (2020), 1113-1136. doi: 10.4208/aamm.OA-2019-0033. [21] S. Li and L. Ling, Weighted least-squares collocation methods for elliptic PDEs with mixed boundary conditions, Engineering Analysis with Boundary Elements, 105 (2019), 146-154.  doi: 10.1016/j.enganabound.2019.04.012. [22] S. Li and L. Ling, Complex pattern formations by spatial varying parameters, Advances in Applied Mathematics and Mechanics, 12 (2020), 1327-1352.  doi: 10.4208/aamm.OA-2018-0266. [23] L. Ling, R. Opfer and R. Schaback, Results on meshless collocation techniques, Engineering Analysis with Boundary Elements, 30 (2006), 247-253.  doi: 10.1016/j.enganabound.2005.08.008. [24] S. Liu and X. Liu, Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries, Discrete & Continuous Dynamical Systems - B, 25 (2020), 141-159.  doi: 10.3934/dcdsb.2019176. [25] A. Madzvamuse, H. S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133. [26] A. Madzvamuse, P. K. Maini and A. J. Wathen, A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains, Journal of Scientific Computing, 24 (2005), 247-262.  doi: 10.1007/s10915-004-4617-7. [27] A. Madzvamuse, A. J. Wathen and P. K. Maini, A moving grid finite element method applied to a model biological pattern generator, Journal of Computational Physics, 190 (2003), 478-500.  doi: 10.1016/S0021-9991(03)00294-8. [28] J. D. Murray, Mathematical biology, vol. 19 of Biomathematics, Springer, Berlin, Germany 1989. doi: 10.1007/978-3-662-08539-4. [29] Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Communications in Computational Physics, 3 (2008), 406-426. [30] Y. Qiu, W. Chen and Q. Nie, A hybrid method for stiff reaction-diffusion equations, Discrete & Continuous Dynamical Systems - B, 24 (2019), 6387-6417.  doi: 10.3934/dcdsb.2019144. [31] S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation, Journal of Mathematical Biology, 34 (1995), 148-176.  doi: 10.1007/BF00178771. [32] J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, Journal of Theoretical Biology, 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0. [33] C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction–diffusion systems on evolving domains, Journal of Mathematical Biology, 64 (2012), 41-67.  doi: 10.1007/s00285-011-0404-x. [34] Z. Xing and L. Wen, The fast implementation of the ADI-CN method for a class of two dimensional Riesz space fractional diffusion equations, Advances in Applied Mathematics and Mechanics, 11 (2019), 942-956. doi: 10.4208/aamm.OA-2018-0162.

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##### References:
 [1] A. Alphonse, C. M. Elliott and B. Stinner, An abstract framework for parabolic PDEs on evolving spaces, Port. Math., 72 (2015), 1–46. doi: 10.4171/PM/1955. [2] S. Bonaccorsi and G. Guatteri, A variational approach to evolution problems with variable domains, Journal of Differential Equations, 175 (2001), 51-70.  doi: 10.1006/jdeq.2000.3959. [3] K. C. Cheung, L. Ling and R. Schaback, $H^2$-Convergence of least-squares kernel collocation methods, SIAM Journal on Numerical Analysis, 56 (2018), 614-633. doi: 10.1137/16M1072863. [4] E. J. Crampin, E. A. Gaffney and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bulletin of Mathematical Biology, 61 (1999), 1093-1120.  doi: 10.1006/bulm.1999.0131. [5] E. J. Crampin, W. W. Hackborn and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bulletin of Mathematical Biology, 64 (2002), 747-769. doi: 10.1006/bulm.2002.0295. [6] M. Dehghan, M. Abbaszadeh and A. Mohebbi, A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model, Engineering Analysis with Boundary Elements, 56 (2015), 129-144.  doi: 10.1016/j.enganabound.2015.02.005. [7] D. Edelmann, Finite element analysis for a diffusion equation on a harmonically evolving domain, preprint, arXiv: 2009.11105. [8] R. I. Fernandes, B. Bialecki and G. Fairweather, An ADI extrapolated Crank–Nicolson orthogonal spline collocation method for nonlinear reaction–diffusion systems on evolving domains, Journal of Computational Physics, 299 (2015), 561-580.  doi: 10.1016/j.jcp.2015.07.016. [9] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234. [10] L. A González, J. C Vanegas and D. A Garzón, Formación de patrones en sistemas de reacción-difusión en dominios crecientes, Revista Internacional de Métodos Numéricos, 25 (2009), 145–161. [11] P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behavior in isothermal reactions, Journal of Physical Chemistry, 89 (1985), 22-32.  doi: 10.1021/j100247a009. [12] Y. C. Hon and R. Schaback, On unsymmetric collocation by radial basis functions, Applied Mathematics and Computation, 119 (2001), 177-186.  doi: 10.1016/S0096-3003(99)00255-6. [13] G. Hu, Z. Qiao and T. Tang, Moving finite element simulations for reaction-diffusion systems, Advances in Applied Mathematics & Mechanics, 4 (2012), 365-381. doi: 10.4208/aamm.10-m11180. [14] E. J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates, Computers & Mathematics with Applications, 19 (1990), 127-145.  doi: 10.1016/0898-1221(90)90270-T. [15] E. J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers & Mathematics with Applications, 19 (1990), 147-161.  doi: 10.1016/0898-1221(90)90271-K. [16] P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, Journal of Differential Equations, 246 (2009), 4702-4730.  doi: 10.1016/j.jde.2008.11.017. [17] S. Kondo and R. Asai, A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0. [18] S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.  doi: 10.1126/science.1179047. [19] O. Lakkis, A. Madzvamuse and C. Venkataraman, Implicit–explicit timestepping with finite element approximation of reaction–diffusion systems on evolving domains, SIAM Journal on Numerical Analysis, 51 (2013), 2309-2330.  doi: 10.1137/120880112. [20] W. Li, K. Rubasinghe, G. Yao and L. H. Kuo, The modified localized method of approximated particular solutions for linear and nonlinear convection-diffusion-reaction PDEs, Advances in Applied Mathematics and Mechanics, 12 (2020), 1113-1136. doi: 10.4208/aamm.OA-2019-0033. [21] S. Li and L. Ling, Weighted least-squares collocation methods for elliptic PDEs with mixed boundary conditions, Engineering Analysis with Boundary Elements, 105 (2019), 146-154.  doi: 10.1016/j.enganabound.2019.04.012. [22] S. Li and L. Ling, Complex pattern formations by spatial varying parameters, Advances in Applied Mathematics and Mechanics, 12 (2020), 1327-1352.  doi: 10.4208/aamm.OA-2018-0266. [23] L. Ling, R. Opfer and R. Schaback, Results on meshless collocation techniques, Engineering Analysis with Boundary Elements, 30 (2006), 247-253.  doi: 10.1016/j.enganabound.2005.08.008. [24] S. Liu and X. Liu, Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries, Discrete & Continuous Dynamical Systems - B, 25 (2020), 141-159.  doi: 10.3934/dcdsb.2019176. [25] A. Madzvamuse, H. S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete and Continuous Dynamical Systems-Series A, 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133. [26] A. Madzvamuse, P. K. Maini and A. J. Wathen, A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains, Journal of Scientific Computing, 24 (2005), 247-262.  doi: 10.1007/s10915-004-4617-7. [27] A. Madzvamuse, A. J. Wathen and P. K. Maini, A moving grid finite element method applied to a model biological pattern generator, Journal of Computational Physics, 190 (2003), 478-500.  doi: 10.1016/S0021-9991(03)00294-8. [28] J. D. Murray, Mathematical biology, vol. 19 of Biomathematics, Springer, Berlin, Germany 1989. doi: 10.1007/978-3-662-08539-4. [29] Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Communications in Computational Physics, 3 (2008), 406-426. [30] Y. Qiu, W. Chen and Q. Nie, A hybrid method for stiff reaction-diffusion equations, Discrete & Continuous Dynamical Systems - B, 24 (2019), 6387-6417.  doi: 10.3934/dcdsb.2019144. [31] S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation, Journal of Mathematical Biology, 34 (1995), 148-176.  doi: 10.1007/BF00178771. [32] J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, Journal of Theoretical Biology, 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0. [33] C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reaction–diffusion systems on evolving domains, Journal of Mathematical Biology, 64 (2012), 41-67.  doi: 10.1007/s00285-011-0404-x. [34] Z. Xing and L. Wen, The fast implementation of the ADI-CN method for a class of two dimensional Riesz space fractional diffusion equations, Advances in Applied Mathematics and Mechanics, 11 (2019), 942-956. doi: 10.4208/aamm.OA-2018-0162.
For the SH model with the unit square domain as reference domain, the $L^2(\Omega)$ error (a) under different overdetermined setting $n_X = k n_Z$ with $\Delta t = 0.01, \ T = 10$; (b) under different kernel smoothness with $k = 1$, $\Delta t = 0.01, \ T = 10$; (c) comparison between forward Euler method and RK2 with parameters $\Delta t = [5E-1, 1E-1, 5E-2, 2E-2]$, $m = 4, \ T = 1$, $n_Z = 55^2$, $k = 1$, $n_X = n_Z$
For the SH model with the unit square domain as reference domain, when $m = 4$, time step $\Delta t = 0.005, \ T = 2001$, the results under fixed discrete sets $n_Z = 25^2$ and different global refinement strategies with $\nu = 1, 2, 3$ in $\rm(17)$, $N_0 = 15^2$ in $\rm(18)$: (a) the $L^2(\Omega)$ error; (b) the number of discrete set $\sqrt{N_t}$; (c) CPU time
In Example 2, when use domain growth function $\rho(t) = \exp(0.001t)$ and parameters $D_u = 1, \ D_v = 10, \ a = 0.1, \ b = 0.9, \ \gamma = 10, \ m = 4, \ \Delta t = 0.01$, the patterns at different time $t$ of SH model under fixed discrete points $n_Z = 30^2$
In Example 2, under same setting with Figure 4, the patterns at different time $t$ of SH model under global refinement with the parameter $\nu = 1$ in $\rm(17)$ and the discrete set increasing from $n_Z = 18^2$ to $n_Z = 35^2$ in $\rm(18)$ as show in Figure 8 (a)
In Example 3, when use global refinement strategy with discrete sets in the domain as in Figure 8 (b) and the domain growth function as $\rho(t) = 1+9\sin(\pi t/1000)$, patterns generated by $D_u = 1, \ D_v = 10, \ a = 0.1, \ b = 0.9, \ \gamma = 10, \ m = 4$ and $\Delta t = 0.005$
In Example 4, in the hexagon domain, when use global refinement strategy with discrete sets in the domain as in Figure 8 (c) and the domain growth function as $\rho(t) = \exp(0.001t)$, patterns generated by $D_u = 1, \ D_v = 10, \ a = 0.1, \ b = 0.9, \ \gamma = 114, \ m = 4$ and $\Delta t = 0.005$
In Example 5, in the star-shape domain, when use global refinement strategy with discrete sets in the domain as in Figure 8 (d) and the domain growth function as $\rho(t) = \exp(0.001t)$, patterns generated by $D_u = 1, \ D_v = 10, \ a = 0.1, \ b = 0.9, \ \gamma = 10, \ m = 4$ and $\Delta t = 0.005$
For the SH model, the change of trial centers $N_t$ or $\sqrt{N_t}$ as number of time steps $n_T$ increases under different global refinement settings for different examples: (a) for EX2, in the square domain the $\nu = 1$ in $\rm(17)$; (b) for EX3, in the square domain, $\nu = 5$ in $\rm(17)$; (c) for EX4, in the hexagon domain, $\nu = 1$ in $\rm(17)$; (d) for EX5, in the star-shape domain, $\nu = 2$ in $\rm(17)$
When $\Delta t = 0.005$, $T = 750$ and $m = 6, \ \epsilon = 5$ in SH model, $L^2(\Omega)$ errors and convergence rates comparison between our scheme and [8,Example 1]
 $N$ $e_h$ error Rate $e_h$ ($r=3$ in [8,Example 1]) Rate $10^2$ $0.807*10^{-3}$ $0.171*10^{-3}$ $20^2$ $0.489*10^{-4}$ $4.042$ $0.106*10^{-4}$ $4.001$ $30^2$ $0.807*10^{-5}$ $4.443$ $0.209*10^{-5}$ $4.002$
 $N$ $e_h$ error Rate $e_h$ ($r=3$ in [8,Example 1]) Rate $10^2$ $0.807*10^{-3}$ $0.171*10^{-3}$ $20^2$ $0.489*10^{-4}$ $4.042$ $0.106*10^{-4}$ $4.001$ $30^2$ $0.807*10^{-5}$ $4.443$ $0.209*10^{-5}$ $4.002$
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