# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021057
##### References:
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Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] J. D. Murray, Mathematical biology, vol. 19 of Biomathematics, Springer, Berlin, Germany 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar [29] Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Communications in Computational Physics, 3 (2008), 406-426.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] D. Edelmann, Finite element analysis for a diffusion equation on a harmonically evolving domain, preprint, arXiv: 2009.11105. Google Scholar [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.  Google Scholar [9] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] J. D. Murray, Mathematical biology, vol. 19 of Biomathematics, Springer, Berlin, Germany 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar [29] Z. Qiao, Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Communications in Computational Physics, 3 (2008), 406-426.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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 $n_Z = 30^2$
, 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)">Figure 4.  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)
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 $\Delta t = 0.005$">Figure 5.  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 $\Delta t = 0.005$
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 $\Delta t = 0.005$">Figure 6.  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 $\Delta t = 0.005$
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 $\Delta t = 0.005$">Figure 7.  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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