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

* Corresponding author: Charles-Edouard Bréhier

Received  January 2018 Revised  October 2018 Published  April 2019

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.

Citation: Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 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. AntonD. 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. BessaihZ. 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.

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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.

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X. ChenD. 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. ChenY. 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.

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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.

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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.

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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. EvansH. 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.

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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.

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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.

[30]

I. GyöngyS. 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. HighamX. 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. HutzenthalerA. 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.

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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. JentzenP. 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. 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.

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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.

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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.

[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. doi: 10.1007/s40072-018-0116-y.

[42]

A. Jentzen, D. Salimova and T. Welti, Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations, Journal of Mathematical Analysis and Applications, 469 (2019), 661–704, arXiv: 1710.07123, 2017. doi: 10.1016/j.jmaa.2018.09.032.

[43]

D. JeongS. LeeD. LeeJ. Shin and J. Kim, Comparison study of numerical methods for solving the Allen–Cahn equation, Computational Materials Science, 111 (2016), 131-136. doi: 10.1016/j.commatsci.2015.09.005.

[44]

M. A. KatsoulakisG. T. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem, Interfaces Free Bound., 9 (2007), 1-30. doi: 10.4171/IFB/154.

[45]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[46]

P. E. Kloeden and S. Shott, Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs, J. Appl. Math. Stochastic Anal., 14 (2001), 47–53. Special issue: Advances in applied stochastics. doi: 10.1155/S1048953301000053.

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R. KohnF. OttoM. G. Reznikoff and E. Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 60 (2007), 393-438. doi: 10.1002/cpa.20144.

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M. KovácsS. Larsson and F. Lindgren, On the backward Euler approximation of the stochastic Allen-Cahn equation, J. Appl. Probab., 52 (2015), 323-338. doi: 10.1239/jap/1437658601.

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M. KovácsS. Larsson and F. Lindgren, On the discretisation in time of the stochastic Allen-Cahn equation, Math. Nachr., 291 (2018), 966-995. doi: 10.1002/mana.201600283.

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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. AntonD. 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. BessaihZ. 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. ChenD. 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. ChenY. 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. EvansH. 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. GyöngyS. 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. HighamX. 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. HutzenthalerA. 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. JentzenP. 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. doi: 10.1007/s40072-018-0116-y.

[42]

A. Jentzen, D. Salimova and T. Welti, Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations, Journal of Mathematical Analysis and Applications, 469 (2019), 661–704, arXiv: 1710.07123, 2017. doi: 10.1016/j.jmaa.2018.09.032.

[43]

D. JeongS. LeeD. LeeJ. Shin and J. Kim, Comparison study of numerical methods for solving the Allen–Cahn equation, Computational Materials Science, 111 (2016), 131-136. doi: 10.1016/j.commatsci.2015.09.005.

[44]

M. A. KatsoulakisG. T. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem, Interfaces Free Bound., 9 (2007), 1-30. doi: 10.4171/IFB/154.

[45]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, volume 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[46]

P. E. Kloeden and S. Shott, Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs, J. Appl. Math. Stochastic Anal., 14 (2001), 47–53. Special issue: Advances in applied stochastics. doi: 10.1155/S1048953301000053.

[47]

R. KohnF. OttoM. G. Reznikoff and E. Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 60 (2007), 393-438. doi: 10.1002/cpa.20144.

[48]

M. KovácsS. Larsson and F. Lindgren, On the backward Euler approximation of the stochastic Allen-Cahn equation, J. Appl. Probab., 52 (2015), 323-338. doi: 10.1239/jap/1437658601.

[49]

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Figure 1.  Mean square error order for $T = 1$, $\Delta x = 2.5~10^{-4}$ and $10^{5}$ independent realizations
Figure 2.  Weak error order for $T = 1$, $\Delta x = 2.5~10^{-4}$ and $10^{5}$ independent realizations
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