Research article Special Issues

A kernel-free boundary integral method for reaction-diffusion equations

  • Received: 18 October 2024 Revised: 03 December 2024 Accepted: 03 January 2025 Published: 08 February 2025
  • This paper was based on a kernel-free boundary integral (KFBI) method for solving the reaction-diffusion equation. The KFBI method serves as a general elliptic solvers for boundary value problems in an irregular problem domain. Unlike traditional boundary integral methods, the KFBI method avoids complicated direct integral calculations. Instead, a Cartesian grid-based five-point compact difference scheme was used to discretize the equivalent simple interface problem, whose solution is the integral involved in the corresponding boundary integral equations (BIEs). The resulting linear system was treated with a fast Fourier transform (FFT)-based elliptic solver, and the BIEs were iteratively solved by the generalized minimal residual (GMRES) method. The first step in solving the reaction-diffusion equation was to discretize the time variable with a two-stage second-order semi-implicit Runge-Kutta (SIRK) method, which transforms the problem into a spatial modified Helmholtz equation in each time step and can be solved by the KFBI method later. The proposed algorithm had second-order accuracy in both time and space even for small diffusion problems, and the computational work was roughly proportional to the number of grid nodes in the Cartesian grid due to the fast elliptic solver used. Numerical results verified the stability, efficiency, and accuracy of the method.

    Citation: Yijun Chen, Yaning Xie. A kernel-free boundary integral method for reaction-diffusion equations[J]. Electronic Research Archive, 2025, 33(2): 556-581. doi: 10.3934/era.2025026

    Related Papers:

  • This paper was based on a kernel-free boundary integral (KFBI) method for solving the reaction-diffusion equation. The KFBI method serves as a general elliptic solvers for boundary value problems in an irregular problem domain. Unlike traditional boundary integral methods, the KFBI method avoids complicated direct integral calculations. Instead, a Cartesian grid-based five-point compact difference scheme was used to discretize the equivalent simple interface problem, whose solution is the integral involved in the corresponding boundary integral equations (BIEs). The resulting linear system was treated with a fast Fourier transform (FFT)-based elliptic solver, and the BIEs were iteratively solved by the generalized minimal residual (GMRES) method. The first step in solving the reaction-diffusion equation was to discretize the time variable with a two-stage second-order semi-implicit Runge-Kutta (SIRK) method, which transforms the problem into a spatial modified Helmholtz equation in each time step and can be solved by the KFBI method later. The proposed algorithm had second-order accuracy in both time and space even for small diffusion problems, and the computational work was roughly proportional to the number of grid nodes in the Cartesian grid due to the fast elliptic solver used. Numerical results verified the stability, efficiency, and accuracy of the method.



    加载中


    [1] R. Bastiaansen, A. Doelman, The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space, Physica D, 388 (2019), 45–72. https://doi.org/10.1016/j.physd.2018.09.003 doi: 10.1016/j.physd.2018.09.003
    [2] M. Chirilus-Bruckner, A. Doelman, P. van Heijster, J. D. M. Rademacher, Butterfly catastrophe for fronts in a three-component reaction–diffusion system, J. Nonlinear Sci., 25 (2015), 87–129. https://doi.org/10.1007/s00332-014-9222-9 doi: 10.1007/s00332-014-9222-9
    [3] D. Gomez, J. Wei, Z. Yang, Multi-spike solutions to the one-dimensional subcritical fractional Schnakenberg system, Physica D, 448 (2023), 133720. https://doi.org/10.1016/j.physd.2023.133720 doi: 10.1016/j.physd.2023.133720
    [4] O. Jaibi, A. Doelman, M. Chirilus-Bruckner, E. Meron, The existence of localized vegetation patterns in a systematically reduced model for dryland vegetation, Physica D, 412 (2020), 132637. https://doi.org/10.1016/j.physd.2020.132637 doi: 10.1016/j.physd.2020.132637
    [5] A. Iuorio, F. Veerman, The influence of autotoxicity on the dynamics of vegetation spots, Physica D, 427 (2021), 133015. https://doi.org/10.1016/j.physd.2021.133015 doi: 10.1016/j.physd.2021.133015
    [6] P. Carter, A. Doelman, K. Lilly, E. Obermayer, S. Rao, Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction–diffusion equations, Physica D, 444 (2023), 133596. https://doi.org/10.1016/j.physd.2022.133596 doi: 10.1016/j.physd.2022.133596
    [7] L. K. Bieniasz, M. Vynnycky, S. McKee, Integral equation-based simulation of transient experiments for an EC2 mechanism: Beyond the steady state simplification, Electrochim. Acta, 428 (2022), 140896. https://doi.org/10.1016/j.electacta.2022.140896 doi: 10.1016/j.electacta.2022.140896
    [8] V. Lucarini, L. Serdukova, G. Margazoglou, Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model, Nonlinear Processes Geophys., 29 (2022), 183–205. https://doi.org/10.5194/npg-29-183-2022 doi: 10.5194/npg-29-183-2022
    [9] A. Gupta, A. Kaushik, M. Sharma, A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with Robin-boundary conditions, Numer. Methods Partial Differ. Equations, 39 (2023), 1220–1250. https://doi.org/10.1002/num.22931 doi: 10.1002/num.22931
    [10] J. Zhang, X. Liu, Convergence and supercloseness in a balanced norm of finite element methods on Bakhvalov-type meshes for reaction–diffusion problems, J. Sci. Comput., 88 (2021), 27. https://doi.org/10.1007/s10915-021-01542-8 doi: 10.1007/s10915-021-01542-8
    [11] A. Kaushik, V. Kumar, M. Sharma, N. Sharma, A modified graded mesh and higher order finite element method for singularly perturbed reaction–diffusion problems, Math. Comput. Simul., 185 (2021), 486–496. https://doi.org/10.1016/j.matcom.2021.01.006 doi: 10.1016/j.matcom.2021.01.006
    [12] S. I. Ei, M. Kuwamura, Y. Morita, A variational approach to singular perturbation problems in reaction–diffusion systems, Physica D, 207 (2005), 171–219. https://doi.org/10.1016/j.physd.2005.05.020 doi: 10.1016/j.physd.2005.05.020
    [13] K. Aarthika, R. Shiromani, V. Shanthi, A higher-order finite difference method for two-dimensional singularly perturbed reaction-diffusion with source-term-discontinuous problem, Comput. Math. Appl., 118 (2022), 56–73. https://doi.org/10.1016/j.camwa.2022.04.016 doi: 10.1016/j.camwa.2022.04.016
    [14] X. Cai, D. L. Cai, R. Q. Wu, K. H. Xie, High accuracy non-equidistant method for singular perturbation reaction-diffusion problem, Appl. Math. Mech., 30 (2009), 175–182. https://doi.org/10.1007/s10483-009-0205-8 doi: 10.1007/s10483-009-0205-8
    [15] S. C. S. Rao, A. K. Chaturvedi, Analysis and implementation of a computational technique for a coupled system of two singularly perturbed parabolic semilinear reaction–diffusion equations having discontinuous source terms, Commun. Nonlinear Sci. Numer. Simul., 108 (2022), 106232. https://doi.org/10.1016/j.cnsns.2021.106232 doi: 10.1016/j.cnsns.2021.106232
    [16] X. Meng, M. Stynes, Balanced-norm and energy-norm error analyses for a backward Euler/FEM solving a singularly perturbed parabolic reaction-diffusion problem, J. Sci. Comput., 92 (2022), 67. https://doi.org/10.1007/s10915-022-01931-7 doi: 10.1007/s10915-022-01931-7
    [17] M. Faustmann, J. M. Melenk, Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction–diffusion problems: Corner domains, Comput. Math. Appl., 74 (2017), 1576–1589. https://doi.org/10.1016/j.camwa.2017.03.015 doi: 10.1016/j.camwa.2017.03.015
    [18] L. P. Franca, A. L. Madureira, F. Valentin, Towards multiscale functions: Enriching finite element spaces with local but not bubble-like functions, Comput. Methods Appl. Mech. Eng., 194 (2005), 3006–3021. https://doi.org/10.1016/j.cma.2004.07.029 doi: 10.1016/j.cma.2004.07.029
    [19] A. Secer, I. Onder, M. Ozisik, Sinc-Galerkin method for solving system of singular perturbed reaction-diffusion problems, Sigma J. Eng. Nat. Sci., 39 (2021), 203–212. https://doi.org/10.14744/sigma.2021.00010 doi: 10.14744/sigma.2021.00010
    [20] R. Lin, A discontinuous Galerkin least-squares finite element method for solving coupled singularly perturbed reaction–diffusion equations, J. Comput. Appl. Math., 307 (2016), 134–142. https://doi.org/10.1016/j.cam.2016.02.052 doi: 10.1016/j.cam.2016.02.052
    [21] X. Ma, J. Zhang, Supercloseness in a balanced norm of the NIPG method on Shishkin mesh for a reaction diffusion problem, Appl. Math. Comput., 444 (2023), 127828. https://doi.org/10.1016/j.amc.2022.127828 doi: 10.1016/j.amc.2022.127828
    [22] T. Valanarasu, N. Ramanujan, Asymptotic initial-value method for singularly-perturbed boundary-value problems for second-order ordinary differential equations, J. Optim. Theory Appl., 116 (2003), 167–182. https://doi.org/10.1023/A:1022118420907 doi: 10.1023/A:1022118420907
    [23] J. Zhang, L. Ge, J. Kouatchou, A two colorable fourth-order compact difference scheme and parallel iterative solution of the 3D convection diffusion equation, Math. Comput. Simul., 54 (2000), 65–80. https://doi.org/10.1016/S0378-4754(00)00205-6 doi: 10.1016/S0378-4754(00)00205-6
    [24] K. Pan, D. He, H. Hu, An extrapolation cascadic multigrid method combined with a fourth-order compact scheme for 3D Poisson equation, J. Sci. Comput., 70 (2017), 1180–1203. https://doi.org/10.1007/s10915-016-0275-9 doi: 10.1007/s10915-016-0275-9
    [25] J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional convection–diffusion equation, Commun. Numer. Methods Eng., 14 (1998), 209–218. https://doi.org/10.1002/(SICI)1099-0887(199803)14:3%3C209::AID-CNM139%3E3.0.CO;2-P doi: 10.1002/(SICI)1099-0887(199803)14:3%3C209::AID-CNM139%3E3.0.CO;2-P
    [26] J. Kouatchou, J. Zhang, Optimal injection operator and high order schemes for multigrid solution of 3D Poisson equation, Int. J. Comput. Math., 76 (2000), 173–190. https://doi.org/10.1080/00207160008805018 doi: 10.1080/00207160008805018
    [27] J. Zhang, Fast and high accuracy multigrid solution of the three dimensional Poisson equation, J. Comput. Phys., 143 (1998), 449–461. https://doi.org/10.1006/jcph.1998.5982 doi: 10.1006/jcph.1998.5982
    [28] M. M. Gupta, J. Zhang, High accuracy multigrid solution of the 3D convection–diffusion equation, Appl. Math. Comput., 113 (2000), 249–274. https://doi.org/10.1016/S0096-3003(99)00085-5 doi: 10.1016/S0096-3003(99)00085-5
    [29] M. M. Gupta, J. Kouatchou, Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation, Numer. Methods Partial Differ. Equations, 14 (1998), 593–606. https://doi.org/10.1002/(SICI)1098-2426(199809)14:5%3C593::AID-NUM4%3E3.0.CO;2-D doi: 10.1002/(SICI)1098-2426(199809)14:5%3C593::AID-NUM4%3E3.0.CO;2-D
    [30] W. F. Spotz, G. F. Carey, A high-order compact formulation for the 3D Poisson equation, Numer. Methods Partial Differ. Equations, 12 (1996), 235–243. https://doi.org/10.1002/(SICI)1098-2426(199603)12:2%3C235::AID-NUM6%3E3.0.CO;2-R doi: 10.1002/(SICI)1098-2426(199603)12:2%3C235::AID-NUM6%3E3.0.CO;2-R
    [31] Y. Kyei, J. P. Roop, G. Tang, A family of sixth-order compact finite-difference schemes for the three-dimensional Poisson equation, Adv. Numer. Anal., 2010 (2010), 352174. https://doi.org/10.1155/2010/352174 doi: 10.1155/2010/352174
    [32] Y. Ge, Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation, J. Comput. Phys., 229 (2010), 6381–6391. https://doi.org/10.1016/j.jcp.2010.04.048 doi: 10.1016/j.jcp.2010.04.048
    [33] Y. Ge, F. Cao, J. Zhang, A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids, J. Comput. Phys., 234 (2013), 199–216. https://doi.org/10.1016/j.jcp.2012.09.034 doi: 10.1016/j.jcp.2012.09.034
    [34] A. Mayo, A. Greenbaum, Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions, J. Comput. Phys., 220 (2007), 900–914. https://doi.org/10.1016/j.jcp.2006.05.042 doi: 10.1016/j.jcp.2006.05.042
    [35] G. Saldanha, Single cell high order difference schemes for Poisson's equation in three variables, Appl. Math. Comput., 186 (2007), 548–557. https://doi.org/10.1016/j.amc.2006.07.126 doi: 10.1016/j.amc.2006.07.126
    [36] R. F. Boisvert, A fourth-order-accurate Fourier method for the Helmholtz equation in three dimensions, ACM Trans. Math. Software, 13 (1987), 221–234. https://doi.org/10.1145/29380.29863 doi: 10.1145/29380.29863
    [37] W. Hackbusch, Z. P. Nowak, On the fast matrix multiplication in the boundary element method by panel clustering, Numer. Math., 54 (1989), 463–491. https://doi.org/10.1007/BF01396324 doi: 10.1007/BF01396324
    [38] Y. Wang, J. Zhang, Fast and robust sixth-order multigrid computation for the three-dimensional convection–diffusion equation, J. Comput. Appl. Math., 234 (2010), 3496–3506. https://doi.org/10.1016/j.cam.2010.05.022 doi: 10.1016/j.cam.2010.05.022
    [39] C. Bacuta, D. Hayes, J. Jacavage, Efficient discretization and preconditioning of the singularly perturbed reaction-diffusion problem, Comput. Math. Appl., 109 (2022), 270–279. https://doi.org/10.1016/j.camwa.2022.01.031 doi: 10.1016/j.camwa.2022.01.031
    [40] H. Chen, Y. Su, B. D. Shizgal, A direct spectral collocation Poisson solver in polar and cylindrical coordinates, J. Comput. Phys., 160 (2000), 453–469. https://doi.org/10.1006/jcph.2000.6461 doi: 10.1006/jcph.2000.6461
    [41] G. Biros, L. Ying, D. Zorin, A fast solver for the Stokes equations with distributed forces in complex geometries, J. Comput. Phys., 193 (2004), 317–348. https://doi.org/10.1016/j.jcp.2003.08.011 doi: 10.1016/j.jcp.2003.08.011
    [42] A. Greenbaum, A. Mayo, Rapid parallel evaluation of integrals in potential theory on general three-dimensional regions, J. Comput. Phys., 145 (1998), 731–742. https://doi.org/10.1006/jcph.1998.6048 doi: 10.1006/jcph.1998.6048
    [43] L. Ying, G. Biros, D. Zorin, A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains, J. Comput. Phys., 219 (2006), 247–275. https://doi.org/10.1016/j.jcp.2006.03.021 doi: 10.1016/j.jcp.2006.03.021
    [44] A. N. Marques, J. C. Nave, R. R. Rosales, A correction function method for Poisson problems with interface jump conditions, J. Comput. Phys., 230 (2011), 7567–7597. https://doi.org/10.1016/j.jcp.2011.06.014 doi: 10.1016/j.jcp.2011.06.014
    [45] J. R. Phillips, J. K. White, A precorrected-FFT method for electrostatic analysis of complicated 3D structures, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst., 16 (1997), 1059–1072. https://doi.org/10.1109/43.662670 doi: 10.1109/43.662670
    [46] W. Ying, C. S. Henriquez, A Kernel-free boundary integral method for elliptic boundary value problems, J. Comput. Phys., 227 (2007), 1046–1074. https://doi.org/10.1016/j.jcp.2007.08.021 doi: 10.1016/j.jcp.2007.08.021
    [47] E. Braverman, M. Israeli, A. Averbuch, L. Vozovoi, A fast 3D Poisson solver of arbitrary order accuracy, J. Comput. Phys., 144 (1998), 109–136. https://doi.org/10.1006/jcph.1998.6001 doi: 10.1006/jcph.1998.6001
    [48] W. Ying, W. C. Wang, A Kernel-free boundary integral method for implicitly defined surfaces, J. Comput. Phys., 252 (2013), 606–624. https://doi.org/10.1016/j.jcp.2013.06.019 doi: 10.1016/j.jcp.2013.06.019
    [49] Y. Xie, W. Ying, A fourth-order Kernel-free boundary integral method for the modified Helmholtz equation, J. Sci. Comput., 78 (2019), 1632–1658. https://doi.org/10.1007/s10915-018-0821-8 doi: 10.1007/s10915-018-0821-8
    [50] A. N. Marques, J. C. Nave, R. R. Rosales, High order solution of Poisson problems with piecewise constant coefficients and interface jumps, J. Comput. Phys., 335 (2017), 497–515. https://doi.org/10.1016/j.jcp.2017.01.029 doi: 10.1016/j.jcp.2017.01.029
    [51] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements, Springer, 2007. https://doi.org/10.1007/978-0-387-68805-3
    [52] A. Frangi, A. di Gioia, Multipole BEM for the evaluation of damping forces on MEMS, Comput. Mech., 37 (2005), 24–31. https://doi.org/10.1007/s00466-005-0694-1 doi: 10.1007/s00466-005-0694-1
    [53] A. A. Samarskii, The Theory of Difference Schemes, CRC Press, 2001. https://doi.org/10.1201/9780203908518
    [54] X. Feng, H. Song, T. Tang, J. Yang, Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation, Inverse Probl. Imaging, 7 (2013), 679–695. https://doi.org/10.3934/ipi.2013.7.679 doi: 10.3934/ipi.2013.7.679
  • Reader Comments
  • © 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(261) PDF downloads(35) Cited by(0)

Article outline

Figures and Tables

Figures(13)  /  Tables(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog