Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Lattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition

Authors: Martin Sauer and Wilhelm Stannat
Journal: Math. Comp. 84 (2015), 743-766
MSC (2010): Primary 60H15, 60H35; Secondary 35R60, 65C30, 92C20
Published electronically: August 12, 2014
MathSciNet review: 3290962
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider strong convergence of the finite differences approximation in space for stochastic reaction diffusion equations in one dimension with multiplicative noise under a one-sided Lipschitz condition only. The equation may be additionally coupled with a noisy control variable with global Lipschitz condition but no diffusion. We derive convergence with an implicit rate depending on the regularity of the exact solution. This can be made explicit if the variational solution has more than its canonical spatial regularity. As an application, spatially extended FitzHugh-Nagumo systems with noise are considered.

References [Enhancements On Off] (What's this?)

  • [1] Stefano Bonaccorsi and Elisa Mastrogiacomo, Analysis of the stochastic FitzHugh-Nagumo system, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), no. 3, 427-446. MR 2446518 (2009f:35174),
  • [2] Erich Carelli and Andreas Prohl, Rates of convergence for discretizations of the stochastic incompressible Navier-Stokes equations, SIAM J. Numer. Anal. 50 (2012), no. 5, 2467-2496. MR 3022227,
  • [3] G. Bard Ermentrout and David H. Terman, Mathematical Foundations of Neuroscience, Interdisciplinary Applied Mathematics, vol. 35, Springer, New York, 2010. MR 2674516 (2012a:92010)
  • [4] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1:445-466, 1961.
  • [5] R. FitzHugh, Mathematical Models of Excitation and Propagation in Nerve.
    In Biological Engineering, McGraw-Hill, New York, 1969.
  • [6] István Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II, Potential Anal. 11 (1999), no. 1, 1-37. MR 1699161 (2000g:60106),
  • [7] István Gyöngy and Annie Millet, On discretization schemes for stochastic evolution equations, Potential Anal. 23 (2005), no. 2, 99-134. MR 2139212 (2006a:60115),
  • [8] István Gyöngy and Annie Millet, Rate of convergence of space time approximations for stochastic evolution equations, Potential Anal. 30 (2009), no. 1, 29-64. MR 2465711 (2010f:60182),
  • [9] Erika Hausenblas, Numerical analysis of semilinear stochastic evolution equations in Banach spaces, J. Comput. Appl. Math. 147 (2002), no. 2, 485-516. MR 1933610 (2003j:35338),
  • [10] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117:500-544, 1952.
  • [11] Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2130, 1563-1576. MR 2795791 (2012g:65012),
  • [12] Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab. 22 (2012), no. 4, 1611-1641. MR 2985171,
  • [13] Arnulf Jentzen, Pathwise numerical approximation of SPDEs with additive noise under non-global Lipschitz coefficients, Potential Anal. 31 (2009), no. 4, 375-404. MR 2551596 (2011c:60202),
  • [14] James Keener and James Sneyd, Mathematical Physiology, Interdisciplinary Applied Mathematics, vol. 8, Springer-Verlag, New York, 1998. MR 1673204 (2000c:92010)
  • [15] Gabriel J. Lord and Jacques Rougemont, A numerical scheme for stochastic PDEs with Gevrey regularity, IMA J. Numer. Anal. 24 (2004), no. 4, 587-604. MR 2094572 (2005d:60102),
  • [16] Di Liu, Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise, Commun. Math. Sci. 1 (2003), no. 2, 361-375. MR 1980481 (2004c:60187)
  • [17] Wei Liu and Michael Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal. 259 (2010), no. 11, 2902-2922. MR 2719279 (2011m:60196),
  • [18] Roger Pettersson and Mikael Signahl, Numerical approximation for a white noise driven SPDE with locally bounded drift, Potential Anal. 22 (2005), no. 4, 375-393. MR 2135265 (2005k:60215),
  • [19] Claudia Prévôt and Michael Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007. MR 2329435 (2009a:60069)
  • [20] Tony Shardlow, Numerical methods for stochastic parabolic PDEs, Numer. Funct. Anal. Optim. 20 (1999), no. 1-2, 121-145. MR 1683281 (2000g:65004),
  • [21] Tony Shardlow, Numerical simulation of stochastic PDEs for excitable media, J. Comput. Appl. Math. 175 (2005), no. 2, 429-446. MR 2108585 (2005k:65017),
  • [22] Hans Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam, 1978. MR 503903 (80i:46032b)
  • [23] Henry C. Tuckwell, Analytical and simulation results for the stochastic spatial FitzHugh-Nagumo model neuron, Neural Comput. 20 (2008), no. 12, 3003-3033. MR 2467646 (2009m:92026),

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 60H15, 60H35, 35R60, 65C30, 92C20

Retrieve articles in all journals with MSC (2010): 60H15, 60H35, 35R60, 65C30, 92C20

Additional Information

Martin Sauer
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany

Wilhelm Stannat
Affiliation: Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany and Bernstein Center for Computational Neuroscience, Philippstr. 13, D-10115 Berlin, Germany

Keywords: Stochastic reaction diffusion equations, finite difference approximation, FitzHugh-Nagumo system
Received by editor(s): January 27, 2013
Received by editor(s) in revised form: July 22, 2013
Published electronically: August 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society