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A hybrid method for stiff reaction–diffusion equations
1. | Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA |
2. | Department of Mathematics, University of California, Riverside, Riverside, CA 92507, USA |
3. | Department of Mathematics, Department of Developmental and Cell Biology, University of California, Irvine, Irvine, CA 92697, USA |
The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction–diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order $\underline {\text{h}} {\text{ybrid}}$ ${\text{I}}\underline {{\text{IF}}} - \underline {\text{E}} {\text{TD}}$ method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction–diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method.
References:
[1] |
D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to turing spatial patterns, Ecology, 83 (2002), 28-34. Google Scholar |
[2] |
G. Beylkin, J. M. Keiser and L. Vozovoi,
A new class of time discretization schemes for the solution of nonlinear PDEs, Journal of Computational Physics, 147 (1998), 362-387.
doi: 10.1006/jcph.1998.6093. |
[3] |
S. Chen and Y.-T. 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. |
[4] |
S. M. Cox and P. C. Matthews,
Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455.
doi: 10.1006/jcph.2002.6995. |
[5] |
P. D. Dale, J. A. Sherratt and P. K. Maini,
Role of fibroblast migration in collagen fiber formation during fetal and adult dermal wound healing, Bulletin of mathematical biology, 59 (1997), 1077-1100.
doi: 10.1007/BF02460102. |
[6] |
Q. Du and W. Zhu,
Stability analysis and application of the exponential time differencing schemes, Journal of Computational Mathematics, 22 (2014), 200-209.
|
[7] |
Q. Du and W. Zhu,
Analysis and applications of the exponential time differencing schemes and their contour integration modifications, BIT Numerical Mathematics, 45 (2005), 307-328.
doi: 10.1007/s10543-005-7141-8. |
[8] |
A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B.-Z. Shilo and N. Barkai,
Robustness of the bmp morphogen gradient in drosophila embryonic patterning, Nature, 419 (2002), 304-308.
doi: 10.1038/nature01061. |
[9] |
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. |
[10] |
M. Garvie and C. Trenchea, Analysis of two generic spatially extended predator-prey models, Nonlinear Anal. Real World Appl. Google Scholar |
[11] |
M. R. Garvie,
Finite-difference schemes for reaction–diffusion equations modeling predator–prey interactions in matlab, Bulletin of mathematical biology, 69 (2007), 931-956.
doi: 10.1007/s11538-006-9062-3. |
[12] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Biological Cybernetics, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[13] |
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. |
[14] |
M. Hochbruck and A. Ostermann,
Explicit exponential Runge–Kutta methods for semilinear parabolic problems, SIAM Journal on Numerical Analysis, 43 (2005), 1069-1090.
doi: 10.1137/040611434. |
[15] |
M. Hochbruck and A. Ostermann,
Exponential Runge–Kutta methods for parabolic problems, Applied Numerical Mathematics, 53 (2005), 323-339.
doi: 10.1016/j.apnum.2004.08.005. |
[16] |
M. Hochbruck and A. Ostermann,
Exponential integrators, Acta Numerica, 19 (2010), 209-286.
doi: 10.1017/S0962492910000048. |
[17] |
E. E. Holmes, M. A. Lewis, J. Banks and R. Veit,
Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17-29.
doi: 10.2307/1939378. |
[18] |
E. Isaacson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. |
[19] |
T. Jiang and Y.-T. 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. |
[20] |
T. Jiang and Y.-T. Zhang,
Krylov single-step implicit integration factor WENO methods for advection–diffusion–reaction equations, Journal of Computational Physics, 311 (2016), 22-44.
doi: 10.1016/j.jcp.2016.01.021. |
[21] |
L. Ju, X. Liu and W. Leng,
Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 1667-1687.
doi: 10.3934/dcdsb.2014.19.1667. |
[22] |
L. Ju, J. Zhang, L. Zhu and Q. Du,
Fast explicit integration factor methods for semilinear parabolic equations, Journal of Scientific Computing, 62 (2015), 431-455.
doi: 10.1007/s10915-014-9862-9. |
[23] |
A.-K. Kassam and L. N. Trefethen,
Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233.
doi: 10.1137/S1064827502410633. |
[24] |
A. Kicheva, P. Pantazis, T. Bollenbach, Y. Kalaidzidis, T. Bittig, F. Jülicher and M. Gonzalez-Gaitan,
Kinetics of morphogen gradient formation, Science, 315 (2007), 521-525.
doi: 10.1126/science.1135774. |
[25] |
J. D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, John Wiley & Sons, Inc., 1991. |
[26] |
A. D. Lander, Q. Nie and F. Y. Wan, Do morphogen gradients arise by diffusion?, Developmental cell, 2 (2002), 785-796. Google Scholar |
[27] |
S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, vol. 45, Springer-Verlag, Berlin, 2003. |
[28] |
X. Liu and Q. Nie,
Compact integration factor methods for complex domains and adaptive mesh refinement, Journal of Computational Physics, 229 (2010), 5692-5706.
doi: 10.1016/j.jcp.2010.04.003. |
[29] |
D. Lu and Y.-T. 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. |
[30] |
D. Lu and Y.-T. Zhang,
Computational complexity study on Krylov integration factor WENO method for high spatial dimension convection–diffusion problems, Journal of Scientific Computing, 73 (2017), 980-1027.
doi: 10.1007/s10915-017-0398-7. |
[31] |
M. Machen and Y.-T. Zhang, Krylov implicit integration factor methods for semilinear fourth-order equations, Mathematics, 5 (2017), 63.
doi: 10.3390/math5040063. |
[32] |
R. E. Mickens,
Nonstandard finite difference schemes for reaction-diffusion equations, Numerical Methods for Partial Differential Equations, 15 (1999), 201-214.
doi: 10.1002/(SICI)1098-2426(199903)15:2<201::AID-NUM5>3.0.CO;2-H. |
[33] |
Q. Nie, F. Y. Wan, Y.-T. Zhang and X. Liu,
Compact integration factor methods in high spatial dimensions, Journal of Computational Physics, 227 (2008), 5238-5255.
doi: 10.1016/j.jcp.2008.01.050. |
[34] |
Q. Nie, Y.-T. 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. V. Petrovskii and H. Malchow,
A minimal model of pattern formation in a prey-predator system, Mathematical and Computer Modelling, 29 (1999), 49-63.
doi: 10.1016/S0895-7177(99)00070-9. |
[36] |
S. V. Petrovskii and H. Malchow,
Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics, Theoretical population biology, 59 (2001), 157-174.
doi: 10.1006/tpbi.2000.1509. |
[37] |
H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications, Springer Series in Synergetics, 18. Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-96807-5. |
[38] |
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. |
[39] |
J. C. Schulze, P. J. Schmid and J. L. Sesterhenn,
Exponential time integration using Krylov subspaces, International Journal for Numerical Methods in Fluids, 60 (2009), 591-609.
doi: 10.1002/fld.1902. |
[40] |
C. Ta, D. Wang and Q. Nie,
An integration factor method for stochastic and stiff reaction–diffusion systems, Journal of Computational Physics, 295 (2015), 505-522.
doi: 10.1016/j.jcp.2015.04.028. |
[41] |
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 33. Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0569-2. |
[42] |
A. M. Turing, The chemical basis of morphogenesis, Bulletin of mathematical biology, 52 (1990), 153-197. Google Scholar |
[43] |
C. Van Loan, Computational Frameworks for the Fast FOurier Transform, vol. 10, SIAM, 1992.
doi: 10.1137/1.9781611970999. |
[44] |
D. Wang, W. Chen and Q. Nie,
Semi-implicit integration factor methods on sparse grids for high-dimensional systems, Journal of Computational Physics, 292 (2015), 43-55.
doi: 10.1016/j.jcp.2015.03.033. |
[45] |
D. Wang, L. Zhang and Q. Nie,
Array-representation integration factor method for high-dimensional systems, Journal of Computational Physics, 258 (2014), 585-600.
doi: 10.1016/j.jcp.2013.11.002. |
[46] |
O. Wartlick, A. Kicheva and M. González-Gaitán, Morphogen gradient formation, Cold Spring Harbor perspectives in biology, 1 (2009), a001255.
doi: 10.1101/cshperspect.a001255. |
[47] | A.-M. Wazwaz, Partial Differential Equations, CRC Press, 2002. Google Scholar |
[48] |
A. Wiegmann, Fast Poisson, fast Helmholtz and fast linear elastostatic solvers on rectangular parallelepipeds, Lawrence Berkeley National Laboratory.
doi: 10.2172/982430. |
[49] |
S. R. Yu, M. Burkhardt, M. Nowak, J. Ries, Z. Petrášek, S. Scholpp, P. Schwille and M. Brand,
Fgf8 morphogen gradient forms by a source-sink mechanism with freely diffusing molecules, Nature, 461 (2009), 533-536.
doi: 10.1038/nature08391. |
[50] |
L. Zhang, A. D. Lander and Q. Nie, A reaction–diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts, BMC Systems Biology, 6 (2012), 93.
doi: 10.1186/1752-0509-6-93. |
[51] |
S. Zhao, J. Ovadia, X. Liu, Y.-T. Zhang and Q. Nie,
Operator splitting implicit integration factor methods for stiff reaction–diffusion–advection systems, Journal of Computational Physics, 230 (2011), 5996-6009.
doi: 10.1016/j.jcp.2011.04.009. |
[52] |
L. Zhu, L. Ju and W. Zhao,
Fast high-order compact exponential time differencing runge–kutta methods for second-order semilinear parabolic equations, Journal of Scientific Computing, 67 (2016), 1043-1065.
doi: 10.1007/s10915-015-0117-1. |
[53] |
Y.-L. Zhu, X. Wu, I.-L. Chern and Z.-Z. Sun, Derivative Securities and Difference Methods, Springer, 2004.
doi: 10.1007/978-1-4757-3938-1. |
show all references
References:
[1] |
D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to turing spatial patterns, Ecology, 83 (2002), 28-34. Google Scholar |
[2] |
G. Beylkin, J. M. Keiser and L. Vozovoi,
A new class of time discretization schemes for the solution of nonlinear PDEs, Journal of Computational Physics, 147 (1998), 362-387.
doi: 10.1006/jcph.1998.6093. |
[3] |
S. Chen and Y.-T. 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. |
[4] |
S. M. Cox and P. C. Matthews,
Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455.
doi: 10.1006/jcph.2002.6995. |
[5] |
P. D. Dale, J. A. Sherratt and P. K. Maini,
Role of fibroblast migration in collagen fiber formation during fetal and adult dermal wound healing, Bulletin of mathematical biology, 59 (1997), 1077-1100.
doi: 10.1007/BF02460102. |
[6] |
Q. Du and W. Zhu,
Stability analysis and application of the exponential time differencing schemes, Journal of Computational Mathematics, 22 (2014), 200-209.
|
[7] |
Q. Du and W. Zhu,
Analysis and applications of the exponential time differencing schemes and their contour integration modifications, BIT Numerical Mathematics, 45 (2005), 307-328.
doi: 10.1007/s10543-005-7141-8. |
[8] |
A. Eldar, R. Dorfman, D. Weiss, H. Ashe, B.-Z. Shilo and N. Barkai,
Robustness of the bmp morphogen gradient in drosophila embryonic patterning, Nature, 419 (2002), 304-308.
doi: 10.1038/nature01061. |
[9] |
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. |
[10] |
M. Garvie and C. Trenchea, Analysis of two generic spatially extended predator-prey models, Nonlinear Anal. Real World Appl. Google Scholar |
[11] |
M. R. Garvie,
Finite-difference schemes for reaction–diffusion equations modeling predator–prey interactions in matlab, Bulletin of mathematical biology, 69 (2007), 931-956.
doi: 10.1007/s11538-006-9062-3. |
[12] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Biological Cybernetics, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[13] |
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. |
[14] |
M. Hochbruck and A. Ostermann,
Explicit exponential Runge–Kutta methods for semilinear parabolic problems, SIAM Journal on Numerical Analysis, 43 (2005), 1069-1090.
doi: 10.1137/040611434. |
[15] |
M. Hochbruck and A. Ostermann,
Exponential Runge–Kutta methods for parabolic problems, Applied Numerical Mathematics, 53 (2005), 323-339.
doi: 10.1016/j.apnum.2004.08.005. |
[16] |
M. Hochbruck and A. Ostermann,
Exponential integrators, Acta Numerica, 19 (2010), 209-286.
doi: 10.1017/S0962492910000048. |
[17] |
E. E. Holmes, M. A. Lewis, J. Banks and R. Veit,
Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75 (1994), 17-29.
doi: 10.2307/1939378. |
[18] |
E. Isaacson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. |
[19] |
T. Jiang and Y.-T. 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. |
[20] |
T. Jiang and Y.-T. Zhang,
Krylov single-step implicit integration factor WENO methods for advection–diffusion–reaction equations, Journal of Computational Physics, 311 (2016), 22-44.
doi: 10.1016/j.jcp.2016.01.021. |
[21] |
L. Ju, X. Liu and W. Leng,
Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations, Discrete & Continuous Dynamical Systems-Series B, 19 (2014), 1667-1687.
doi: 10.3934/dcdsb.2014.19.1667. |
[22] |
L. Ju, J. Zhang, L. Zhu and Q. Du,
Fast explicit integration factor methods for semilinear parabolic equations, Journal of Scientific Computing, 62 (2015), 431-455.
doi: 10.1007/s10915-014-9862-9. |
[23] |
A.-K. Kassam and L. N. Trefethen,
Fourth-order time-stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233.
doi: 10.1137/S1064827502410633. |
[24] |
A. Kicheva, P. Pantazis, T. Bollenbach, Y. Kalaidzidis, T. Bittig, F. Jülicher and M. Gonzalez-Gaitan,
Kinetics of morphogen gradient formation, Science, 315 (2007), 521-525.
doi: 10.1126/science.1135774. |
[25] |
J. D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, John Wiley & Sons, Inc., 1991. |
[26] |
A. D. Lander, Q. Nie and F. Y. Wan, Do morphogen gradients arise by diffusion?, Developmental cell, 2 (2002), 785-796. Google Scholar |
[27] |
S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods, vol. 45, Springer-Verlag, Berlin, 2003. |
[28] |
X. Liu and Q. Nie,
Compact integration factor methods for complex domains and adaptive mesh refinement, Journal of Computational Physics, 229 (2010), 5692-5706.
doi: 10.1016/j.jcp.2010.04.003. |
[29] |
D. Lu and Y.-T. 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. |
[30] |
D. Lu and Y.-T. Zhang,
Computational complexity study on Krylov integration factor WENO method for high spatial dimension convection–diffusion problems, Journal of Scientific Computing, 73 (2017), 980-1027.
doi: 10.1007/s10915-017-0398-7. |
[31] |
M. Machen and Y.-T. Zhang, Krylov implicit integration factor methods for semilinear fourth-order equations, Mathematics, 5 (2017), 63.
doi: 10.3390/math5040063. |
[32] |
R. E. Mickens,
Nonstandard finite difference schemes for reaction-diffusion equations, Numerical Methods for Partial Differential Equations, 15 (1999), 201-214.
doi: 10.1002/(SICI)1098-2426(199903)15:2<201::AID-NUM5>3.0.CO;2-H. |
[33] |
Q. Nie, F. Y. Wan, Y.-T. Zhang and X. Liu,
Compact integration factor methods in high spatial dimensions, Journal of Computational Physics, 227 (2008), 5238-5255.
doi: 10.1016/j.jcp.2008.01.050. |
[34] |
Q. Nie, Y.-T. 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. V. Petrovskii and H. Malchow,
A minimal model of pattern formation in a prey-predator system, Mathematical and Computer Modelling, 29 (1999), 49-63.
doi: 10.1016/S0895-7177(99)00070-9. |
[36] |
S. V. Petrovskii and H. Malchow,
Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics, Theoretical population biology, 59 (2001), 157-174.
doi: 10.1006/tpbi.2000.1509. |
[37] |
H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications, Springer Series in Synergetics, 18. Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-96807-5. |
[38] |
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. |
[39] |
J. C. Schulze, P. J. Schmid and J. L. Sesterhenn,
Exponential time integration using Krylov subspaces, International Journal for Numerical Methods in Fluids, 60 (2009), 591-609.
doi: 10.1002/fld.1902. |
[40] |
C. Ta, D. Wang and Q. Nie,
An integration factor method for stochastic and stiff reaction–diffusion systems, Journal of Computational Physics, 295 (2015), 505-522.
doi: 10.1016/j.jcp.2015.04.028. |
[41] |
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 33. Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-0569-2. |
[42] |
A. M. Turing, The chemical basis of morphogenesis, Bulletin of mathematical biology, 52 (1990), 153-197. Google Scholar |
[43] |
C. Van Loan, Computational Frameworks for the Fast FOurier Transform, vol. 10, SIAM, 1992.
doi: 10.1137/1.9781611970999. |
[44] |
D. Wang, W. Chen and Q. Nie,
Semi-implicit integration factor methods on sparse grids for high-dimensional systems, Journal of Computational Physics, 292 (2015), 43-55.
doi: 10.1016/j.jcp.2015.03.033. |
[45] |
D. Wang, L. Zhang and Q. Nie,
Array-representation integration factor method for high-dimensional systems, Journal of Computational Physics, 258 (2014), 585-600.
doi: 10.1016/j.jcp.2013.11.002. |
[46] |
O. Wartlick, A. Kicheva and M. González-Gaitán, Morphogen gradient formation, Cold Spring Harbor perspectives in biology, 1 (2009), a001255.
doi: 10.1101/cshperspect.a001255. |
[47] | A.-M. Wazwaz, Partial Differential Equations, CRC Press, 2002. Google Scholar |
[48] |
A. Wiegmann, Fast Poisson, fast Helmholtz and fast linear elastostatic solvers on rectangular parallelepipeds, Lawrence Berkeley National Laboratory.
doi: 10.2172/982430. |
[49] |
S. R. Yu, M. Burkhardt, M. Nowak, J. Ries, Z. Petrášek, S. Scholpp, P. Schwille and M. Brand,
Fgf8 morphogen gradient forms by a source-sink mechanism with freely diffusing molecules, Nature, 461 (2009), 533-536.
doi: 10.1038/nature08391. |
[50] |
L. Zhang, A. D. Lander and Q. Nie, A reaction–diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts, BMC Systems Biology, 6 (2012), 93.
doi: 10.1186/1752-0509-6-93. |
[51] |
S. Zhao, J. Ovadia, X. Liu, Y.-T. Zhang and Q. Nie,
Operator splitting implicit integration factor methods for stiff reaction–diffusion–advection systems, Journal of Computational Physics, 230 (2011), 5996-6009.
doi: 10.1016/j.jcp.2011.04.009. |
[52] |
L. Zhu, L. Ju and W. Zhao,
Fast high-order compact exponential time differencing runge–kutta methods for second-order semilinear parabolic equations, Journal of Scientific Computing, 67 (2016), 1043-1065.
doi: 10.1007/s10915-015-0117-1. |
[53] |
Y.-L. Zhu, X. Wu, I.-L. Chern and Z.-Z. Sun, Derivative Securities and Difference Methods, Springer, 2004.
doi: 10.1007/978-1-4757-3938-1. |



Method | Reaction |
Truncation error |
IIF2 | ||
iETD2 | ||
hIFE2 | equivalent to iETD2 | |
equivalent to IIF2 | ||
Method | Reaction |
Truncation error |
IIF2 | ||
iETD2 | ||
hIFE2 | equivalent to iETD2 | |
equivalent to IIF2 | ||
![]() |
32 | 64 | 128 | 256 | 512 | 1024 |
1 | ||||||
5 | ||||||
![]() |
32 | 64 | 128 | 256 | 512 | 1024 |
1 | ||||||
5 | ||||||
Neumann | Dirichlet | Mixed | |
BCs | |||
Neumann | Dirichlet | Mixed | |
BCs | |||
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
IIF2 | 8 | 0.00228 | - | 0.09 | 0.05 | 0.04 |
16 | 0.000591 | 1.95 | 0.04 | 0.02 | 0.02 | |
32 | 0.000198 | 1.58 | 0.07 | 0.03 | 0.04 | |
64 | 7.81e-05 | 1.34 | 0.13 | 0.04 | 0.09 | |
128 | 0.000108 | 0.54 | 0.07 | 0.47 | ||
256 | 5.18e-05 | 1.06 | 1.26 | 0.23 | 1.03 | |
512 | 1.83e-05 | 1.50 | 4.00 | 1.39 | 2.61 | |
1024 | 2.07e-05 | 28.30 | 7.75 | 20.55 | ||
2048 | 1.07e-05 | 0.96 | 168.12 | 42.10 | 126.02 | |
4096 | 5.35e-06 | 1.00 | 1148.42 | 265.35 | 883.07 | |
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
iETD2 | 8 | 0.00216 | - | 0.07 | 0.04 | 0.03 |
16 | 0.000539 | 2.00 | 0.07 | 0.04 | 0.03 | |
32 | 0.000135 | 2.00 | 0.12 | 0.06 | 0.06 | |
64 | 3.37e-05 | 2.00 | 0.80 | 0.07 | 0.73 | |
128 | 8.41e-06 | 2.00 | 3.78 | 0.16 | 3.62 | |
256 | 2.1e-06 | 2.00 | 22.99 | 0.54 | 22.45 | |
512 | 5.26e-07 | 2.00 | 289.66 | 2.70 | 286.96 | |
1024 | 1.32e-07 | 2.00 | 2841.66 | 14.65 | 2827.01 | |
2048 | 3.31e-08 | 1.99 | 35348.32 | 91.84 | 35256.48 | |
4096 | - | - | - | - | ||
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
hIFE2 | 8 | 0.00217 | - | 0.12 | 0.09 | 0.03 |
16 | 0.000544 | 1.99 | 0.06 | 0.04 | 0.02 | |
32 | 0.000137 | 1.99 | 0.08 | 0.05 | 0.03 | |
64 | 3.42e-05 | 2.00 | 0.16 | 0.08 | 0.08 | |
128 | 8.75e-06 | 1.97 | 0.76 | 0.17 | 0.59 | |
256 | 2.21e-06 | 1.99 | 1.85 | 0.54 | 1.31 | |
512 | 5.53e-07 | 2.00 | 9.17 | 2.61 | 6.56 | |
1024 | 1.49e-07 | 1.89 | 61.82 | 14.20 | 47.62 | |
2048 | 3.93e-08 | 1.93 | 419.24 | 89.49 | 329.75 | |
4096 | 1.12e-08 | 1.81 | 3096.23 | 603.04 | 2493.19 | |
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
fEIF2 | 8 | 0.00216 | - | 0.37 | 0.37 | 0.00 |
16 | 0.00054 | 2.00 | 0.04 | 0.04 | 0.00 | |
32 | 0.000135 | 2.00 | 0.07 | 0.07 | 0.00 | |
64 | 3.38e-05 | 2.00 | 0.09 | 0.08 | 0.01 | |
128 | 8.44e-06 | 2.00 | 0.54 | 0.18 | 0.36 | |
256 | 2.11e-06 | 2.00 | 1.41 | 0.69 | 0.72 | |
512 | 5.28e-07 | 2.00 | 11.62 | 3.01 | 8.61 | |
1024 | 1.32e-07 | 2.00 | 84.11 | 16.11 | 68.00 | |
2048 | 3.31e-08 | 1.99 | 613.91 | 101.12 | 512.79 | |
4096 | 8.89e-09 | 1.90 | 4700.11 | 707.64 | 3992.47 |
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
IIF2 | 8 | 0.00228 | - | 0.09 | 0.05 | 0.04 |
16 | 0.000591 | 1.95 | 0.04 | 0.02 | 0.02 | |
32 | 0.000198 | 1.58 | 0.07 | 0.03 | 0.04 | |
64 | 7.81e-05 | 1.34 | 0.13 | 0.04 | 0.09 | |
128 | 0.000108 | 0.54 | 0.07 | 0.47 | ||
256 | 5.18e-05 | 1.06 | 1.26 | 0.23 | 1.03 | |
512 | 1.83e-05 | 1.50 | 4.00 | 1.39 | 2.61 | |
1024 | 2.07e-05 | 28.30 | 7.75 | 20.55 | ||
2048 | 1.07e-05 | 0.96 | 168.12 | 42.10 | 126.02 | |
4096 | 5.35e-06 | 1.00 | 1148.42 | 265.35 | 883.07 | |
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
iETD2 | 8 | 0.00216 | - | 0.07 | 0.04 | 0.03 |
16 | 0.000539 | 2.00 | 0.07 | 0.04 | 0.03 | |
32 | 0.000135 | 2.00 | 0.12 | 0.06 | 0.06 | |
64 | 3.37e-05 | 2.00 | 0.80 | 0.07 | 0.73 | |
128 | 8.41e-06 | 2.00 | 3.78 | 0.16 | 3.62 | |
256 | 2.1e-06 | 2.00 | 22.99 | 0.54 | 22.45 | |
512 | 5.26e-07 | 2.00 | 289.66 | 2.70 | 286.96 | |
1024 | 1.32e-07 | 2.00 | 2841.66 | 14.65 | 2827.01 | |
2048 | 3.31e-08 | 1.99 | 35348.32 | 91.84 | 35256.48 | |
4096 | - | - | - | - | ||
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
hIFE2 | 8 | 0.00217 | - | 0.12 | 0.09 | 0.03 |
16 | 0.000544 | 1.99 | 0.06 | 0.04 | 0.02 | |
32 | 0.000137 | 1.99 | 0.08 | 0.05 | 0.03 | |
64 | 3.42e-05 | 2.00 | 0.16 | 0.08 | 0.08 | |
128 | 8.75e-06 | 1.97 | 0.76 | 0.17 | 0.59 | |
256 | 2.21e-06 | 1.99 | 1.85 | 0.54 | 1.31 | |
512 | 5.53e-07 | 2.00 | 9.17 | 2.61 | 6.56 | |
1024 | 1.49e-07 | 1.89 | 61.82 | 14.20 | 47.62 | |
2048 | 3.93e-08 | 1.93 | 419.24 | 89.49 | 329.75 | |
4096 | 1.12e-08 | 1.81 | 3096.23 | 603.04 | 2493.19 | |
Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
fEIF2 | 8 | 0.00216 | - | 0.37 | 0.37 | 0.00 |
16 | 0.00054 | 2.00 | 0.04 | 0.04 | 0.00 | |
32 | 0.000135 | 2.00 | 0.07 | 0.07 | 0.00 | |
64 | 3.38e-05 | 2.00 | 0.09 | 0.08 | 0.01 | |
128 | 8.44e-06 | 2.00 | 0.54 | 0.18 | 0.36 | |
256 | 2.11e-06 | 2.00 | 1.41 | 0.69 | 0.72 | |
512 | 5.28e-07 | 2.00 | 11.62 | 3.01 | 8.61 | |
1024 | 1.32e-07 | 2.00 | 84.11 | 16.11 | 68.00 | |
2048 | 3.31e-08 | 1.99 | 613.91 | 101.12 | 512.79 | |
4096 | 8.89e-09 | 1.90 | 4700.11 | 707.64 | 3992.47 |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
IIF2 | 20 | 10 | 0.00381 | - | 5.32 | 5.25 | 0.07 |
40 | 4.81 | 0.00182 | 1.06 | 5.07 | 4.91 | 0.16 | |
80 | 2.32 | 0.000881 | 1.05 | 5.09 | 4.78 | 0.31 | |
160 | 1.12 | 0.000425 | 1.05 | 5.07 | 4.44 | 0.63 | |
320 | 0.534 | 0.000203 | 1.07 | 5.24 | 3.90 | 1.34 | |
640 | 0.251 | 9.51e-05 | 1.09 | 5.92 | 3.40 | 2.52 | |
1280 | 0.115 | 4.34e-05 | 1.13 | 7.90 | 2.92 | 4.98 | |
2560 | 0.0503 | 1.91e-05 | 1.19 | 12.84 | 2.55 | 10.29 | |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
iETD2 | 20 | 3.99 | 0.00151 | - | 19.63 | 10.88 | 8.75 |
40 | 0.994 | 0.000377 | 2.00 | 28.60 | 10.80 | 17.80 | |
80 | 0.248 | 9.41e-05 | 2.00 | 46.92 | 10.76 | 36.16 | |
160 | 0.0617 | 2.34e-05 | 2.01 | 80.10 | 10.41 | 69.69 | |
320 | 0.0152 | 5.76e-06 | 2.02 | 148.60 | 9.80 | 138.80 | |
640 | 0.00366 | 1.39e-06 | 2.05 | 285.20 | 9.27 | 275.93 | |
1280 | 0.000872 | 3.31e-07 | 2.07 | 567.11 | 8.94 | 558.17 | |
2560 | 0.000227 | 8.61e-08 | 1.94 | 1140.59 | 8.49 | 1132.10 | |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
hIFE2 | 20 | 4.19 | 0.000397 | - | 11.11 | 0.00 | 0.28 |
40 | 1.05 | 0.000397 | 2.00 | 11.96 | 11.39 | 0.57 | |
80 | 0.261 | 9.91e-05 | 2.00 | 11.61 | 10.70 | 0.91 | |
160 | 0.0652 | 2.47e-05 | 2.00 | 12.36 | 10.37 | 1.99 | |
320 | 0.0162 | 6.14e-06 | 2.01 | 13.90 | 9.84 | 4.06 | |
640 | 0.00397 | 1.51e-06 | 2.03 | 17.65 | 9.43 | 8.22 | |
1280 | 0.000971 | 3.68e-07 | 2.03 | 25.08 | 8.88 | 16.20 | |
2560 | 0.000256 | 9.72e-08 | 1.92 | 40.83 | 8.45 | 32.38 | |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
fEIF2 | 20 | 1.49e+29 | 4.43e+25 | - | 12.42 | 11.96 | 0.46 |
40 | 2.9e+48 | 8.65e+44 | 12.61 | 11.78 | 0.83 | ||
80 | 6.04e+73 | 1.8e+70 | 13.07 | 11.46 | 1.61 | ||
160 | 2.27e+96 | 6.77e+92 | 14.43 | 11.20 | 3.23 | ||
320 | 1.92e+79 | 5.71e+75 | 56.72 | 17.20 | 10.59 | 6.61 | |
640 | 0.251 | 7.48e-05 | 265.37 | 23.47 | 9.93 | 13.54 | |
1280 | 0.119 | 3.54e-05 | 1.08 | 35.96 | 9.57 | 26.39 | |
2560 | 0.0603 | 1.8e-05 | 0.98 | 62.05 | 9.08 | 52.97 |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
IIF2 | 20 | 10 | 0.00381 | - | 5.32 | 5.25 | 0.07 |
40 | 4.81 | 0.00182 | 1.06 | 5.07 | 4.91 | 0.16 | |
80 | 2.32 | 0.000881 | 1.05 | 5.09 | 4.78 | 0.31 | |
160 | 1.12 | 0.000425 | 1.05 | 5.07 | 4.44 | 0.63 | |
320 | 0.534 | 0.000203 | 1.07 | 5.24 | 3.90 | 1.34 | |
640 | 0.251 | 9.51e-05 | 1.09 | 5.92 | 3.40 | 2.52 | |
1280 | 0.115 | 4.34e-05 | 1.13 | 7.90 | 2.92 | 4.98 | |
2560 | 0.0503 | 1.91e-05 | 1.19 | 12.84 | 2.55 | 10.29 | |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
iETD2 | 20 | 3.99 | 0.00151 | - | 19.63 | 10.88 | 8.75 |
40 | 0.994 | 0.000377 | 2.00 | 28.60 | 10.80 | 17.80 | |
80 | 0.248 | 9.41e-05 | 2.00 | 46.92 | 10.76 | 36.16 | |
160 | 0.0617 | 2.34e-05 | 2.01 | 80.10 | 10.41 | 69.69 | |
320 | 0.0152 | 5.76e-06 | 2.02 | 148.60 | 9.80 | 138.80 | |
640 | 0.00366 | 1.39e-06 | 2.05 | 285.20 | 9.27 | 275.93 | |
1280 | 0.000872 | 3.31e-07 | 2.07 | 567.11 | 8.94 | 558.17 | |
2560 | 0.000227 | 8.61e-08 | 1.94 | 1140.59 | 8.49 | 1132.10 | |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
hIFE2 | 20 | 4.19 | 0.000397 | - | 11.11 | 0.00 | 0.28 |
40 | 1.05 | 0.000397 | 2.00 | 11.96 | 11.39 | 0.57 | |
80 | 0.261 | 9.91e-05 | 2.00 | 11.61 | 10.70 | 0.91 | |
160 | 0.0652 | 2.47e-05 | 2.00 | 12.36 | 10.37 | 1.99 | |
320 | 0.0162 | 6.14e-06 | 2.01 | 13.90 | 9.84 | 4.06 | |
640 | 0.00397 | 1.51e-06 | 2.03 | 17.65 | 9.43 | 8.22 | |
1280 | 0.000971 | 3.68e-07 | 2.03 | 25.08 | 8.88 | 16.20 | |
2560 | 0.000256 | 9.72e-08 | 1.92 | 40.83 | 8.45 | 32.38 | |
Relative error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |||
fEIF2 | 20 | 1.49e+29 | 4.43e+25 | - | 12.42 | 11.96 | 0.46 |
40 | 2.9e+48 | 8.65e+44 | 12.61 | 11.78 | 0.83 | ||
80 | 6.04e+73 | 1.8e+70 | 13.07 | 11.46 | 1.61 | ||
160 | 2.27e+96 | 6.77e+92 | 14.43 | 11.20 | 3.23 | ||
320 | 1.92e+79 | 5.71e+75 | 56.72 | 17.20 | 10.59 | 6.61 | |
640 | 0.251 | 7.48e-05 | 265.37 | 23.47 | 9.93 | 13.54 | |
1280 | 0.119 | 3.54e-05 | 1.08 | 35.96 | 9.57 | 26.39 | |
2560 | 0.0603 | 1.8e-05 | 0.98 | 62.05 | 9.08 | 52.97 |
(A) | Decomposition (48) | Decomposition (49) | |||
A priori error | Order | A priori error | Order | ||
16 | 0.00103 | - | 0.00102 | - | |
32 | 0.000532 | 0.95 | 0.000255 | 2.00 | |
64 | 0.000328 | 0.70 | 6.37e-05 | 2.00 | |
128 | 8.21e-05 | 2.00 | 1.59e-05 | 2.00 | |
256 | 0.000196 | 3.98e-06 | 2.00 | ||
512 | 0.000106 | 0.88 | 9.95e-07 | 2.00 | |
1024 | 8.69e-06 | 3.61 | 2.49e-07 | 2.00 | |
2048 | 3.37e-05 | 6.19e-08 | 2.01 | ||
4096 | 1.75e-05 | 0.95 | 1.42e-08 | 2.12 | |
(B) | Decomposition (51) | Decomposition (52) | |||
N | A priori error | Order | A priori error | Order | |
16 | 0.00979 | - | 0.00977 | - | |
32 | 0.00245 | 2.00 | 0.00245 | 1.99 | |
64 | 6.67e-04 | 1.88 | 0.000614 | 2.00 | |
128 | 1.67e-04 | 2.00 | 0.000153 | 2.00 | |
256 | 3.94e-04 | -1.24 | 3.83e-05 | 2.00 | |
512 | 2.13e-04 | 0.89 | 9.59e-06 | 2.00 | |
1024 | 1.76e-05 | 3.60 | 2.4e-06 | 2.00 | |
2048 | 6.75e-05 | -1.94 | 5.99e-07 | 2.00 | |
4096 | 3.50e-05 | 0.95 | 1.48e-07 | 2.02 |
(A) | Decomposition (48) | Decomposition (49) | |||
A priori error | Order | A priori error | Order | ||
16 | 0.00103 | - | 0.00102 | - | |
32 | 0.000532 | 0.95 | 0.000255 | 2.00 | |
64 | 0.000328 | 0.70 | 6.37e-05 | 2.00 | |
128 | 8.21e-05 | 2.00 | 1.59e-05 | 2.00 | |
256 | 0.000196 | 3.98e-06 | 2.00 | ||
512 | 0.000106 | 0.88 | 9.95e-07 | 2.00 | |
1024 | 8.69e-06 | 3.61 | 2.49e-07 | 2.00 | |
2048 | 3.37e-05 | 6.19e-08 | 2.01 | ||
4096 | 1.75e-05 | 0.95 | 1.42e-08 | 2.12 | |
(B) | Decomposition (51) | Decomposition (52) | |||
N | A priori error | Order | A priori error | Order | |
16 | 0.00979 | - | 0.00977 | - | |
32 | 0.00245 | 2.00 | 0.00245 | 1.99 | |
64 | 6.67e-04 | 1.88 | 0.000614 | 2.00 | |
128 | 1.67e-04 | 2.00 | 0.000153 | 2.00 | |
256 | 3.94e-04 | -1.24 | 3.83e-05 | 2.00 | |
512 | 2.13e-04 | 0.89 | 9.59e-06 | 2.00 | |
1024 | 1.76e-05 | 3.60 | 2.4e-06 | 2.00 | |
2048 | 6.75e-05 | -1.94 | 5.99e-07 | 2.00 | |
4096 | 3.50e-05 | 0.95 | 1.48e-07 | 2.02 |
Order | ||
1.33e-03 | - | |
3.28e-04 | 2.02 | |
8.17e-05 | 2.01 | |
2.04e-05 | 2.00 | |
5.10e-05 | 2.00 | |
1.27e-06 | 2.00 |
Order | ||
1.33e-03 | - | |
3.28e-04 | 2.02 | |
8.17e-05 | 2.01 | |
2.04e-05 | 2.00 | |
5.10e-05 | 2.00 | |
1.27e-06 | 2.00 |
Method | A-stability | ||
Time-dependent reactions | Nonhomogeneous BCs | ||
IIF2 | Yes | ||
iETD2 | Yes | - | |
fEIF2 | No | ||
hIFE2 | Yes | ||
hIFE2 (transformed) | Yes |
Method | A-stability | ||
Time-dependent reactions | Nonhomogeneous BCs | ||
IIF2 | Yes | ||
iETD2 | Yes | - | |
fEIF2 | No | ||
hIFE2 | Yes | ||
hIFE2 (transformed) | Yes |
Operations per iteration | Total complexity (ratio) | |
IIF2 | 1 | |
iETD2 | ||
hIFE2 | 3 | |
fEIF2 | 5 |
Operations per iteration | Total complexity (ratio) | |
IIF2 | 1 | |
iETD2 | ||
hIFE2 | 3 | |
fEIF2 | 5 |
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