Figure 1.
Mean square error order for $T = 1$ , $\Delta x = 2.5~10^{-4}$ and $10^{5}$ independent realizations
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August
2019, 24(8): 4169-4190.
doi: 10.3934/dcdsb.2019077
Analysis of some splitting schemes for the stochastic Allen-Cahn equation
| 1. | Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne, France |
| 2. | Université Paris-Saclay, CNRS - FR3487, Fédération de Mathématiques de CentraleSupélec, CentraleSupélec, 3 rue Joliot Curie, F-91190 Gif-sur-Yvette, France |
We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution.
We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, $L^2(\Omega)$-convergence of order almost $1/4$, localized on an event of arbitrarily large probability, then convergence in probability of order almost $1/4$.
The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.
Mathematics Subject Classification:
Primary: 65C30, 60H35; Secondary: 60H15.
Citation:
Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B,
2019, 24
(8)
: 4169-4190.
doi: 10.3934/dcdsb.2019077
![]()
References:
| [1] |
S. Allen and J. Cahn,
A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal. Mater., 27 (1979), 1085-1095.
|
| [2] |
R. Anton, D. Cohen and L. Quer-Sardanyons,
A fully discrete approximation of the one-dimensional stochastic heat equation, to appear, IMA J. Numer. Anal., 36 (2016), 400-420.
doi: 10.1093/imanum/dry060. |
| [3] |
S. Becker, B. Gess, A. Jentzen and P. E. Kloeden, Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations, arXiv: 1711.02423, 2017. |
| [4] |
S. Becker and A. Jentzen, Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations, Stochastic Processes and their Applications, 129 (2019), 28–69, arXiv: 1601.05756.
doi: 10.1016/j.spa.2018.02.008. |
| [5] |
H. Bessaih, Z. Brzeźniak and A. Millet,
Splitting up method for the 2D stochastic Navier-Stokes equations, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 433-470.
doi: 10.1007/s40072-014-0041-7. |
| [6] |
H. Bessaih and A. Millet, On strong L2 convergence of numerical schemes for the stochastic 2d Navier-Stokes equations, arXiv: 1801.03548, 2018. |
| [7] |
C.-E. Bréhier and A. Debussche,
Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient, Journal de Mathématiques Pures et Appliquées, 129 (2018), 193-254.
doi: 10.1016/j.matpur.2018.08.010. |
| [8] |
C.-E. Bréhier, M. Gazeau, L. Goudenège, and M. Rousset, Analysis and simulation of rare events for SPDEs, In CEMRACS 2013–-Modelling and Simulation of Complex Systems: Stochastic and Deterministic Approaches, volume 48 of ESAIM Proc. Surveys, pages 364–384. EDP Sci., Les Ulis, 2015.
doi: 10.1051/proc/201448017. |
| [9] |
S. Cerrai, Second Order PDE's in Finite and Infinite Dimension, volume 1762 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2001. A probabilistic approach.
doi: 10.1007/b80743. |
| [10] |
X. Chen, D. Hilhorst and E. Logak,
Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term, Nonlinear Anal., 28 (1997), 1283-1298.
doi: 10.1016/S0362-546X(97)82875-1. |
| [11] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
| [12] |
D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, Ann. Appl. Probab., 29 (2019), 653–716, arXiv: 1408.1108.
doi: 10.1214/17-AAP1352. |
| [13] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 152 of
Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge,
second edition, 2014.
doi: 10.1017/CBO9781107295513. |
| [14] |
A. M. Davie and J. G. Gaines,
Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comp., 70 (2001), 121-134.
doi: 10.1090/S0025-5718-00-01224-2. |
| [15] |
A. Debussche,
Weak approximation of stochastic partial differential equations: The nonlinear case, Math. Comp., 80 (2011), 89-117.
doi: 10.1090/S0025-5718-2010-02395-6. |
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L. C. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
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L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
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L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅱ, Trans. Amer. Math. Soc., 330 (1992), 321-332.
doi: 10.1090/S0002-9947-1992-1068927-8. |
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W. G. Faris and G. Jona-Lasinio,
Large fluctuations for a nonlinear heat equation with noise, J. Phys. A, 15 (1982), 3025-3055.
doi: 10.1088/0305-4470/15/10/011. |
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T. Funaki,
The scaling limit for a stochastic PDE and the separation of phases, Probab. Theory Related Fields, 102 (1995), 221-288.
doi: 10.1007/BF01213390. |
| [21] |
T. Funaki,
Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sin. (Engl. Ser.), 15 (1999), 407-438.
doi: 10.1007/BF02650735. |
| [22] |
I. Gyöngy,
Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. Ⅰ, Potential Anal., 9 (1998), 1-25.
doi: 10.1023/A:1008615012377. |
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I. Gyöngy,
Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. Ⅱ, Potential Anal., 11 (1999), 1-37.
doi: 10.1023/A:1008699504438. |
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I. Gyöngy and N. Krylov, On the rate of convergence of splitting-up approximations for SPDEs, In Stochastic Inequalities and Applications, volume 56 of Progr. Probab., pages 301–321. Birkhäuser, Basel, 2003. |
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I. Gyöngy and N. Krylov,
On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591.
doi: 10.1214/aop/1048516528. |
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I. Gyöngy and A. Millet,
On discretization schemes for stochastic evolution equations, Potential Anal., 23 (2005), 99-134.
doi: 10.1007/s11118-004-5393-6. |
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I. Gyöngy and A. Millet, Rate of convergence of implicit approximations for stochastic evolution equations, In Stochastic Differential Equations: Theory and Applications, volume 2 of Interdiscip. Math. Sci., pages 281–310. World Sci. Publ., Hackensack, NJ, 2007.
doi: 10.1142/9789812770639_0011. |
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I. Gyöngy and D. Nualart,
Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise, Stochastic Process. Appl., 58 (1995), 57-72.
doi: 10.1016/0304-4149(95)00010-5. |
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I. Gyöngy and D. Nualart,
Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Potential Anal., 7 (1997), 725-757.
doi: 10.1023/A:1017998901460. |
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I. Gyöngy, S. Sabanis and D. Šiška,
Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stoch. Partial Differ. Equ. Anal. Comput., 4 (2016), 225-245.
doi: 10.1007/s40072-015-0057-7. |
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D. J. Higham, X. Mao and A. M. Stuart,
Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
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M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015), v+99pp.
doi: 10.1090/memo/1112. |
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Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.
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A. Jentzen,
Higher order pathwise numerical approximations of SPDEs with additive noise, SIAM J. Numer. Anal., 49 (2011), 642-667.
doi: 10.1137/080740714. |
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A. Jentzen, P. Kloeden and G. Winkel,
Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab., 21 (2011), 908-950.
doi: 10.1214/10-AAP711. |
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A. Jentzen and P. E. Kloeden,
Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 649-667.
doi: 10.1098/rspa.2008.0325. |
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A. Jentzen and P. Pušnik, Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities, arXiv: 1504.03523, 2015. |
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show all references
References:
| [1] |
S. Allen and J. Cahn,
A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metal. Mater., 27 (1979), 1085-1095.
|
| [2] |
R. Anton, D. Cohen and L. Quer-Sardanyons,
A fully discrete approximation of the one-dimensional stochastic heat equation, to appear, IMA J. Numer. Anal., 36 (2016), 400-420.
doi: 10.1093/imanum/dry060. |
| [3] |
S. Becker, B. Gess, A. Jentzen and P. E. Kloeden, Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations, arXiv: 1711.02423, 2017. |
| [4] |
S. Becker and A. Jentzen, Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations, Stochastic Processes and their Applications, 129 (2019), 28–69, arXiv: 1601.05756.
doi: 10.1016/j.spa.2018.02.008. |
| [5] |
H. Bessaih, Z. Brzeźniak and A. Millet,
Splitting up method for the 2D stochastic Navier-Stokes equations, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 433-470.
doi: 10.1007/s40072-014-0041-7. |
| [6] |
H. Bessaih and A. Millet, On strong L2 convergence of numerical schemes for the stochastic 2d Navier-Stokes equations, arXiv: 1801.03548, 2018. |
| [7] |
C.-E. Bréhier and A. Debussche,
Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient, Journal de Mathématiques Pures et Appliquées, 129 (2018), 193-254.
doi: 10.1016/j.matpur.2018.08.010. |
| [8] |
C.-E. Bréhier, M. Gazeau, L. Goudenège, and M. Rousset, Analysis and simulation of rare events for SPDEs, In CEMRACS 2013–-Modelling and Simulation of Complex Systems: Stochastic and Deterministic Approaches, volume 48 of ESAIM Proc. Surveys, pages 364–384. EDP Sci., Les Ulis, 2015.
doi: 10.1051/proc/201448017. |
| [9] |
S. Cerrai, Second Order PDE's in Finite and Infinite Dimension, volume 1762 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2001. A probabilistic approach.
doi: 10.1007/b80743. |
| [10] |
X. Chen, D. Hilhorst and E. Logak,
Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term, Nonlinear Anal., 28 (1997), 1283-1298.
doi: 10.1016/S0362-546X(97)82875-1. |
| [11] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
| [12] |
D. Conus, A. Jentzen and R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, Ann. Appl. Probab., 29 (2019), 653–716, arXiv: 1408.1108.
doi: 10.1214/17-AAP1352. |
| [13] |
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, volume 152 of
Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge,
second edition, 2014.
doi: 10.1017/CBO9781107295513. |
| [14] |
A. M. Davie and J. G. Gaines,
Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations, Math. Comp., 70 (2001), 121-134.
doi: 10.1090/S0025-5718-00-01224-2. |
| [15] |
A. Debussche,
Weak approximation of stochastic partial differential equations: The nonlinear case, Math. Comp., 80 (2011), 89-117.
doi: 10.1090/S0025-5718-2010-02395-6. |
| [16] |
L. C. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
| [17] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
| [18] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature. Ⅱ, Trans. Amer. Math. Soc., 330 (1992), 321-332.
doi: 10.1090/S0002-9947-1992-1068927-8. |
| [19] |
W. G. Faris and G. Jona-Lasinio,
Large fluctuations for a nonlinear heat equation with noise, J. Phys. A, 15 (1982), 3025-3055.
doi: 10.1088/0305-4470/15/10/011. |
| [20] |
T. Funaki,
The scaling limit for a stochastic PDE and the separation of phases, Probab. Theory Related Fields, 102 (1995), 221-288.
doi: 10.1007/BF01213390. |
| [21] |
T. Funaki,
Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sin. (Engl. Ser.), 15 (1999), 407-438.
doi: 10.1007/BF02650735. |
| [22] |
I. Gyöngy,
Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. Ⅰ, Potential Anal., 9 (1998), 1-25.
doi: 10.1023/A:1008615012377. |
| [23] |
I. Gyöngy,
Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. Ⅱ, Potential Anal., 11 (1999), 1-37.
doi: 10.1023/A:1008699504438. |
| [24] |
I. Gyöngy and N. Krylov, On the rate of convergence of splitting-up approximations for SPDEs, In Stochastic Inequalities and Applications, volume 56 of Progr. Probab., pages 301–321. Birkhäuser, Basel, 2003. |
| [25] |
I. Gyöngy and N. Krylov,
On the splitting-up method and stochastic partial differential equations, Ann. Probab., 31 (2003), 564-591.
doi: 10.1214/aop/1048516528. |
| [26] |
I. Gyöngy and A. Millet,
On discretization schemes for stochastic evolution equations, Potential Anal., 23 (2005), 99-134.
doi: 10.1007/s11118-004-5393-6. |
| [27] |
I. Gyöngy and A. Millet, Rate of convergence of implicit approximations for stochastic evolution equations, In Stochastic Differential Equations: Theory and Applications, volume 2 of Interdiscip. Math. Sci., pages 281–310. World Sci. Publ., Hackensack, NJ, 2007.
doi: 10.1142/9789812770639_0011. |
| [28] |
I. Gyöngy and D. Nualart,
Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise, Stochastic Process. Appl., 58 (1995), 57-72.
doi: 10.1016/0304-4149(95)00010-5. |
| [29] |
I. Gyöngy and D. Nualart,
Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise, Potential Anal., 7 (1997), 725-757.
doi: 10.1023/A:1017998901460. |
| [30] |
I. Gyöngy, S. Sabanis and D. Šiška,
Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stoch. Partial Differ. Equ. Anal. Comput., 4 (2016), 225-245.
doi: 10.1007/s40072-015-0057-7. |
| [31] |
D. J. Higham, X. Mao and A. M. Stuart,
Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.
doi: 10.1137/S0036142901389530. |
| [32] |
M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015), v+99pp.
doi: 10.1090/memo/1112. |
| [33] |
M. Hutzenthaler, A. Jentzen and P. E. Kloeden,
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.
doi: 10.1214/11-AAP803. |
| [34] |
M. Hutzenthaler, A. Jentzen and D. Salimova, Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations, arXiv: 1604.02053, 2016. |
| [35] |
A. Jentzen,
Higher order pathwise numerical approximations of SPDEs with additive noise, SIAM J. Numer. Anal., 49 (2011), 642-667.
doi: 10.1137/080740714. |
| [36] |
A. Jentzen, P. Kloeden and G. Winkel,
Efficient simulation of nonlinear parabolic SPDEs with additive noise, Ann. Appl. Probab., 21 (2011), 908-950.
doi: 10.1214/10-AAP711. |
| [37] |
A. Jentzen and P. E. Kloeden,
Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 649-667.
doi: 10.1098/rspa.2008.0325. |
| [38] |
A. Jentzen and P. E. Kloeden, Taylor Approximations for Stochastic Partial Differential Equations, volume 83 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011.
doi: 10.1137/1.9781611972016. |
| [39] |
A. Jentzen and R. Kurniawan, Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients, arXiv: 1501.03539, 2015. |
| [40] |
A. Jentzen and P. Pušnik, Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities, arXiv: 1504.03523, 2015. |
| [41] |
A. Jentzen and P. Pušnik, Exponential moments for numerical approximations of stochastic partial differential equations, Stochastics and Partial Differential Equations: Analysis and Computations, 6 (2018), 565–617, arXiv: 1609.07031, 2016.
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Figure 2.
Weak error order for $T = 1$ , $\Delta x = 2.5~10^{-4}$ and $10^{5}$ independent realizations
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