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

  • * Corresponding author: Xinfeng Liu

    * Corresponding author: Xinfeng Liu
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  • 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.

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

    Citation:

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

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

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

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

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV
  • [1] W. BaoY. DuZ. Lin and H. Zhu, Free boundary models for mosquito range movement driven by climate warming, Journal of Mathematical Biology, 76 (2018), 841-875.  doi: 10.1007/s00285-017-1159-9.
    [2] G. BuntingY. Du and K. Krakowski, Spreading speed revisited: analysis of a free boundary model, Networks and Heterogeneous Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.
    [3] K. Burrage and J. C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM Journal on Numerical Analysis, 16 (1979), 46-57.  doi: 10.1137/0716004.
    [4] L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, American Mathematical Soc., 2005. doi: 10.1090/gsm/068.
    [5] Y. CaoA. Faghri and W. S. Chang, A numerical analysis of Stefan problems for generalized multi-dimensional phase-change structures using the enthalpy transforming model, International Journal of Heat and Mass Transfer, 32 (1989), 1289-1298.  doi: 10.1016/0017-9310(89)90029-X.
    [6] H. ChenC. Min and F. Gibou, A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate, Journal of Computational Physics, 228 (2009), 5803-5818.  doi: 10.1016/j.jcp.2009.04.044.
    [7] S. ChenB. MerrimanS. Osher and P. Smereka, A simple level set method for solving Stefan problems, Journal of Computational Physics, 135 (1997), 8-29.  doi: 10.1006/jcph.1997.5721.
    [8] S. Chen and Y. Zhang, Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: application to discontinuous Galerkin methods, Journal of Computational Physics, 230 (2011), 4336-4352.  doi: 10.1016/j.jcp.2011.01.010.
    [9] I. L. ChernJ. GlimmO. McBryanB. Plohr and S. Yaniv, Front tracking for gas dynamics, Journal of Computational Physics, 62 (1986), 83-110.  doi: 10.1016/0021-9991(86)90101-4.
    [10] J. CrankFree and Moving Boundary Problems, Clarendon Press, Oxford, 1984. 
    [11] Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, Journal of Differential Equations, 253 (2012), 996-1035.  doi: 10.1016/j.jde.2012.04.014.
    [12] Y. Du and Z. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM Journal on Mathematical Analysis, 42 (2010), 377-405.  doi: 10.1137/090771089.
    [13] Y. Du and B. Lou, Spreading and vanishing in nonlinear diffusion problems with free boundaries, Journal of the European Mathematical Society, 17 (2015), 2673-2724.  doi: 10.4171/JEMS/568.
    [14] Y. DuH. Matano and K. Wang, Regularity and asymptotic behavior of nonlinear Stefan problems, Archive for Rational Mechanics and Analysis, 212 (2014), 957-1010.  doi: 10.1007/s00205-013-0710-0.
    [15] R. Fedkiw and S. Osher, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, 153. Springer-Verlag, New York, 2003. doi: 10.1007/b98879.
    [16] E. Gallopoulos and Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods, SIAM Journal on Scientific and Statistical Computing, 13 (1992), 1236-1264.  doi: 10.1137/0913071.
    [17] F. Gibou and R. Fedkiw, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, Journal of Computational Physics, 202 (2005), 577-601.  doi: 10.1016/j.jcp.2004.07.018.
    [18] J. GlimmX. L. LiY. Liu and N. Zhao, Conservative front tracking and level set algorithms, Proceedings of the National Academy of Sciences, 98 (2001), 14198-14201.  doi: 10.1073/pnas.251420998.
    [19] E. Hairer and G. Wanner, Stiff differential equations solved by Radau methods, Journal of Computational and Applied Mathematics, 111 (1999), 93-111.  doi: 10.1016/S0377-0427(99)00134-X.
    [20] N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM Journal on Matrix Analysis and Applications, 26 (2005), 1179-1193.  doi: 10.1137/04061101X.
    [21] J. Hilditch and P. Colella, A front tracking method for compressible flames in one dimension, SIAM Journal on Scientific Computing, 16 (1995), 755-772.  doi: 10.1137/0916045.
    [22] M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM Journal on Numerical Analysis, 34 (1997), 1911-1925.  doi: 10.1137/S0036142995280572.
    [23] J. HuaJ. F. Stene and P. Lin, Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method, Journal of Computational Physics, 227 (2008), 3358-3382.  doi: 10.1016/j.jcp.2007.12.002.
    [24] T. Jiang and Y. Zhang, Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations, Journal of Computational Physics, 253 (2013), 368-388.  doi: 10.1016/j.jcp.2013.07.015.
    [25] T. Jiang and Y. Zhang, Krylov single-step implicit integration factor WENO method for advection-diffusion-reaction equations, Journal of Computational Physics, 311 (2016), 22-44.  doi: 10.1016/j.jcp.2016.01.021.
    [26] H. G. Landau, Heat conduction in a melting solid, Quarterly of Applied Mathematics, 8 (1950), 81-94.  doi: 10.1090/qam/33441.
    [27] R. J. Leveque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analysis, 31 (1994), 1019-1044.  doi: 10.1137/0731054.
    [28] R. J. Leveque and K. M. Shyue, Two-dimensional front tracking based on high resolution wave propagation methods, Journal of Computational Physics, 123 (1996), 354-368.  doi: 10.1006/jcph.1996.0029.
    [29] S. Liu and X. Liu, Numerical methods for a two-species competition-diffusion model with free boundaries, Mathematics, 6 (2018), 72. doi: 10.3390/math6050072.
    [30] D. Lu and Y. Zhang, Krylov integration factor method on sparse grids for high spatial dimension convection-diffusion equations, Journal of Scientific Computing, 69 (2016), 736-763.  doi: 10.1007/s10915-016-0216-7.
    [31] M. M. Mac Low and R. S. Klessen, Control of star formation by supersonic turbulence, Reviews of Modern Physics, 76 (2004), 125.
    [32] C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Review, 45 (2003), 3-49.  doi: 10.1137/S00361445024180.
    [33] Q. NieF. Y. WanY. Zhang and X. Liu, Compact integration factor methods in high spatial dimensions, Journal of Computational Physics, 277 (2008), 5238-5255.  doi: 10.1016/j.jcp.2008.01.050.
    [34] Q. NieY. Zhang and R. Zhao, Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214 (2006), 521-537.  doi: 10.1016/j.jcp.2005.09.030.
    [35] S. Osher and R. P. Fedkiw, Level set methods: an overview and some recent results, Journal of Computational Physics, 169 (2001), 463-502.  doi: 10.1006/jcph.2000.6636.
    [36] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.
    [37] D. PengB. MerrimanS. OsherH. Zhao and M. Kang, A PDE-based fast local level set method, Journal of Computational Physics, 155 (1999), 410-438.  doi: 10.1006/jcph.1999.6345.
    [38] C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.
    [39] M. A. PiquerasR. Company and L. Jodar, A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model, Journal of Computational and Applied Mathematics, 309 (2017), 473-481.  doi: 10.1016/j.cam.2016.02.029.
    [40] L. I. Rubinstein, The Stefan Problem, Providence, RI: American Mathematical Society, 1971.
    [41] Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM Journal on Numerical Analysis, 29 (1992), 209-228.  doi: 10.1137/0729014.
    [42] J. A. Sethian, A fast marching level set method for monotonically advancing fronts, Proceedings of the National Academy of Sciences, 93 (1996), 1591-1595.  doi: 10.1073/pnas.93.4.1591.
    [43] J. A. SethianLevel Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999. 
    [44] G. D. SmithNumerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985. 
    [45] M. SussmanP. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114 (1994), 146-159. 
    [46] L. N. Trefethen and D. Bau, III Numerical Linear Algebra, SIAM, 1997. doi: 10.1137/1.9780898719574.
    [47] S. O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, Journal of Computational Physics, 100 (1992), 25-37. 
    [48] A. Wiegmann and K. P. Bube, The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analysis, 35 (1998), 177-200.  doi: 10.1137/S003614299529378X.
    [49] J. J. XuZ. LiJ. Lowengrub and H. Zhao, A level-set method for interfacial flows with surfactant, Journal of Computational Physics, 212 (2006), 590-616.  doi: 10.1016/j.jcp.2005.07.016.
    [50] H. K. ZhaoT. ChanB. Merriman and S. Osher, A variational level set approach to multiphase motion, Journal of Computational Physics, 127 (1996), 179-195.  doi: 10.1006/jcph.1996.0167.
    [51] L. Zhu and C. S. Peskin, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, Journal of Computational Physics, 179 (2002), 452-468.  doi: 10.1006/jcph.2002.7066.
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