October  2019, 24(10): 5337-5354. doi: 10.3934/dcdsb.2019061

Numerical solution of partial differential equations with stochastic Neumann boundary conditions

Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, Iran

Received  May 2018 Revised  November 2018 Published  April 2019

The aim of this paper is to study the numerical solution of partial differential equations with boundary forcing. For spatial discretization we apply the Galerkin method and for time discretization we will use a method based on the accelerated exponential Euler method. Our purpose is to investigate the convergence of the proposed method, but the main difficulty in carrying out this construction is that at the forced boundary the solution is expected to be unbounded. Therefore the error estimates are performed in the $ L_p $ spaces.

Citation: Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061
References:
[1]

A. Abdulle and G. A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys., 231 (2012), 2482-2497.  doi: 10.1016/j.jcp.2011.11.039.  Google Scholar

[2]

D. Blömker and A. Jentzen, Galerkin approximations for the stochastic burgers equation, SIAM J. Numer. Anal., 51 (2013), 694-715.  doi: 10.1137/110845756.  Google Scholar

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D. BlömkerM. Kamrani and S. M. Hosseini, Full discretization of Stochastic Burgers Equation with correlated noise, IMA J. Numer. Anal., 33 (2013), 825-848.  doi: 10.1093/imanum/drs035.  Google Scholar

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Z. BrzeniakB. GoldysS. Peszat and F. Russo, Second order PDEs with Dirichlet white noise boundary conditions, J Evol Equ., 15 (2015), 1-26.  doi: 10.1007/s00028-014-0246-2.  Google Scholar

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S. Cerrai and M. Freidlin, Fast Transport Asymptotics for stochastic RDEs with Boundary noise, Ann. Prob., 39 (2011), 369-405.  doi: 10.1214/10-AOP552.  Google Scholar

[6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar
[7]

G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochast Stochast Rep., 42 (1993), 167-182.  doi: 10.1080/17442509308833817.  Google Scholar

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., 44, Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
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D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Commun. Part. Diff. Eq., 27 (2002), 1283-1299.  doi: 10.1081/PDE-120005839.  Google Scholar

[10]

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.  Google Scholar

[11]

A. Ichikawa, Stability of parabolic equations with boundary and pointwise noise, Stochastic Differential Systems Filtering and Control, Lecture Notes in Control and Information Sciences, 69 (1985), 55-66.  doi: 10.1007/BFb0005059.  Google Scholar

[12]

M. Kamrani and D. Blömker, Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition, J Comput Appl Math., 323 (2017), 123-135.  doi: 10.1016/j.cam.2017.04.012.  Google Scholar

[13]

P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. Math., 10 (2007), 235-253.  doi: 10.1112/S1461157000001388.  Google Scholar

[14]

L. Lapidus and N. Amunds, On Chemical Reactor Theory, Prentice-Hall, 1977. Google Scholar

[15]

S. Maire and É. Tanré, Monte Carlo approximations of the Neumann problem, Monte Carlo Methods and Applications, De Gruyter, 19 (2013), 201-236.  doi: 10.1515/mcma-2013-0010.  Google Scholar

[16]

B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm, Sup. Pisa Cl. Sci., 22 (1995), 55-93.   Google Scholar

[17]

W. W. Mohammed and D. Blömker, Fast diffusion limit for reaction-diffusion systems with stochastic neumann boundary conditions, SIAM J. Math. Anal., 48 (2016), 3547-3578.  doi: 10.1137/140981952.  Google Scholar

[18]

R. Schnaubelt and M. Veraar, Stochastic equations with boundary noise, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 609-629.  doi: 10.1007/978-3-0348-0075-4_30.  Google Scholar

[19]

R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Prob., 22 (1994), 2071-2121.  doi: 10.1214/aop/1176988495.  Google Scholar

[20]

R. B. Sowers, New Asymptotic Results for Stochastic Partial Differential Equations, Ph.D dissertation, University of Maryland. Google Scholar

[21]

R. Vold and M. Vold, Colloid and Interface Chemistry, Addison-Wesley, 1983. Google Scholar

[22]

W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations, IMA J. Appl. Math., 78 (2009), 1237-1264.  doi: 10.1093/imamat/hxs019.  Google Scholar

[23]

E. WeinanD. Liu and E. vanden Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure App. Math., 58 (2005), 1544-1585.  doi: 10.1002/cpa.20088.  Google Scholar

[24]

S. Xu and J. Duan, A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions, Appl. Math. Comput., 217 (2011), 9532-9542.  doi: 10.1016/j.amc.2011.03.137.  Google Scholar

show all references

References:
[1]

A. Abdulle and G. A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comput. Phys., 231 (2012), 2482-2497.  doi: 10.1016/j.jcp.2011.11.039.  Google Scholar

[2]

D. Blömker and A. Jentzen, Galerkin approximations for the stochastic burgers equation, SIAM J. Numer. Anal., 51 (2013), 694-715.  doi: 10.1137/110845756.  Google Scholar

[3]

D. BlömkerM. Kamrani and S. M. Hosseini, Full discretization of Stochastic Burgers Equation with correlated noise, IMA J. Numer. Anal., 33 (2013), 825-848.  doi: 10.1093/imanum/drs035.  Google Scholar

[4]

Z. BrzeniakB. GoldysS. Peszat and F. Russo, Second order PDEs with Dirichlet white noise boundary conditions, J Evol Equ., 15 (2015), 1-26.  doi: 10.1007/s00028-014-0246-2.  Google Scholar

[5]

S. Cerrai and M. Freidlin, Fast Transport Asymptotics for stochastic RDEs with Boundary noise, Ann. Prob., 39 (2011), 369-405.  doi: 10.1214/10-AOP552.  Google Scholar

[6] G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series, 229, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511662829.  Google Scholar
[7]

G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochast Stochast Rep., 42 (1993), 167-182.  doi: 10.1080/17442509308833817.  Google Scholar

[8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl., 44, Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[9]

D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary, Commun. Part. Diff. Eq., 27 (2002), 1283-1299.  doi: 10.1081/PDE-120005839.  Google Scholar

[10]

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.  Google Scholar

[11]

A. Ichikawa, Stability of parabolic equations with boundary and pointwise noise, Stochastic Differential Systems Filtering and Control, Lecture Notes in Control and Information Sciences, 69 (1985), 55-66.  doi: 10.1007/BFb0005059.  Google Scholar

[12]

M. Kamrani and D. Blömker, Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition, J Comput Appl Math., 323 (2017), 123-135.  doi: 10.1016/j.cam.2017.04.012.  Google Scholar

[13]

P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations, LMS J. Comput. Math., 10 (2007), 235-253.  doi: 10.1112/S1461157000001388.  Google Scholar

[14]

L. Lapidus and N. Amunds, On Chemical Reactor Theory, Prentice-Hall, 1977. Google Scholar

[15]

S. Maire and É. Tanré, Monte Carlo approximations of the Neumann problem, Monte Carlo Methods and Applications, De Gruyter, 19 (2013), 201-236.  doi: 10.1515/mcma-2013-0010.  Google Scholar

[16]

B. Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm, Sup. Pisa Cl. Sci., 22 (1995), 55-93.   Google Scholar

[17]

W. W. Mohammed and D. Blömker, Fast diffusion limit for reaction-diffusion systems with stochastic neumann boundary conditions, SIAM J. Math. Anal., 48 (2016), 3547-3578.  doi: 10.1137/140981952.  Google Scholar

[18]

R. Schnaubelt and M. Veraar, Stochastic equations with boundary noise, Progress in Nonlinear Differential Equations and Their Applications, 80 (2011), 609-629.  doi: 10.1007/978-3-0348-0075-4_30.  Google Scholar

[19]

R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Prob., 22 (1994), 2071-2121.  doi: 10.1214/aop/1176988495.  Google Scholar

[20]

R. B. Sowers, New Asymptotic Results for Stochastic Partial Differential Equations, Ph.D dissertation, University of Maryland. Google Scholar

[21]

R. Vold and M. Vold, Colloid and Interface Chemistry, Addison-Wesley, 1983. Google Scholar

[22]

W. Wang and A. J. Roberts, Macroscopic reduction for stochastic reaction-diffusion equations, IMA J. Appl. Math., 78 (2009), 1237-1264.  doi: 10.1093/imamat/hxs019.  Google Scholar

[23]

E. WeinanD. Liu and E. vanden Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure App. Math., 58 (2005), 1544-1585.  doi: 10.1002/cpa.20088.  Google Scholar

[24]

S. Xu and J. Duan, A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions, Appl. Math. Comput., 217 (2011), 9532-9542.  doi: 10.1016/j.amc.2011.03.137.  Google Scholar

Figure 1.  Numerical solution of Example 1, for $ N = 32 $, $ \epsilon = 0.01 $ and $ T = \frac{3}{20} $, $ \Delta t = 10^{-4} $
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