doi: 10.3934/dcdsb.2019176

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

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

* Corresponding author: Xinfeng Liu

Received  November 2018 Revised  February 2019 Published  July 2019

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.

Citation: Shuang Liu, Xinfeng Liu. Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019176
References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, American Mathematical Soc., 2005. doi: 10.1090/gsm/068. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10] J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

H. G. Landau, Heat conduction in a melting solid, Quarterly of Applied Mathematics, 8 (1950), 81-94. doi: 10.1090/qam/33441. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

M. M. Mac Low and R. S. Klessen, Control of star formation by supersonic turbulence, Reviews of Modern Physics, 76 (2004), 125.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517. doi: 10.1017/S0962492902000077. Google Scholar

[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. Google Scholar

[40]

L. I. Rubinstein, The Stefan Problem, Providence, RI: American Mathematical Society, 1971. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43] J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999. Google Scholar
[44] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985. Google Scholar
[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. Google Scholar

[46]

L. N. Trefethen and D. Bau, III Numerical Linear Algebra, SIAM, 1997. doi: 10.1137/1.9780898719574. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

L. A. Caffarelli and S. Salsa, A Geometric Approach to Free Boundary Problems, American Mathematical Soc., 2005. doi: 10.1090/gsm/068. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10] J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

H. G. Landau, Heat conduction in a melting solid, Quarterly of Applied Mathematics, 8 (1950), 81-94. doi: 10.1090/qam/33441. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

M. M. Mac Low and R. S. Klessen, Control of star formation by supersonic turbulence, Reviews of Modern Physics, 76 (2004), 125.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

C. S. Peskin, The immersed boundary method, Acta Numerica, 11 (2002), 479-517. doi: 10.1017/S0962492902000077. Google Scholar

[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. Google Scholar

[40]

L. I. Rubinstein, The Stefan Problem, Providence, RI: American Mathematical Society, 1971. Google Scholar

[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. Google Scholar

[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. Google Scholar

[43] J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999. Google Scholar
[44] G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985. Google Scholar
[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. Google Scholar

[46]

L. N. Trefethen and D. Bau, III Numerical Linear Algebra, SIAM, 1997. doi: 10.1137/1.9780898719574. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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
$ 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
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
$ 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
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
$ 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
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
$ 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
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
$ \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
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
$ \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
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
$ \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
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