Skip to main content
SpringerLink
Log in
Book cover

Numerical Methods for Stochastic Partial Differential Equations with White Noise pp 53–97Cite as

Numerical methods for stochastic differential equations

  • Chapter
  • First Online:

  • 3322 Accesses

Part of the book series: Applied Mathematical Sciences ((AMS,volume 196))

Abstract

In this chapter, we discuss some basic aspects of stochastic differential equations (SDEs) including stochastic ordinary (SODEs) and partial differential equations (SPDEs).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   69.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Here we need to verify that \(\int _{0}^{t}\sqrt{X(s)}\,dW(s)\) is indeed Ito’s integral with a square-integrable integrand, by showing that \(\int _{0}^{t}\mathbb{E}[\left \vert X(s)\right \vert ]\,ds <\infty\). See Remark 3.1.7.

References

  1. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, Mineola, 1972). 10th printing, with corrections

    Google Scholar 

  2. I.A. Adamu, G.J. Lord, Numerical approximation of multiplicative SPDEs. Int. J. Comput. Math. 89, 2603–2621 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Alabert, I. Gyöngy, On numerical approximation of stochastic Burgers’ equation, in From Stochastic Calculus to Mathematical Finance (Springer, Berlin, 2006), pp. 1–15

    Book  MATH  Google Scholar 

  4. E.J. Allen, S.J. Novosel, Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations. Stoch. Stoch. Rep. 64, 117–142 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. V.V. Anh, W. Grecksch, A. Wadewitz, A splitting method for a stochastic Goursat problem. Stoch. Anal. Appl. 17, 315–326 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley-Interscience, New York, 1974)

    MATH  Google Scholar 

  7. A. Ashyralyev, M. Akat, An approximation of stochastic hyperbolic equations: case with Wiener process. Math. Methods Appl. Sci. 36, 1095–1106 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Babuska, R. Tempone, G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. V. Bally, A. Millet, M. Sanz-Solé, Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23, 178–222 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. X. Bardina, M. Jolis, L. Quer-Sardanyons, Weak convergence for the stochastic heat equation driven by Gaussian white noise. Electron. J. Probab. 15(39), 1267–1295 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Barth, A. Lang, Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises. Appl. Math. Optim. 66, 387–413 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Barth, A. Lang, Simulation of stochastic partial differential equations using finite element methods. Stochastics 84, 217–231 (2012)

    MathSciNet  MATH  Google Scholar 

  13. A. Barth, A. Lang, L p and almost sure convergence of a Milstein scheme for stochastic partial differential equations. Stoch. Process. Appl. 123, 1563–1587 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Barton-Smith, A. Debussche, L. Di Menza, Numerical study of two-dimensional stochastic NLS equations. Numer. Methods Partial Differ. Equ. 21, 810–842 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Bauzet, On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments. J. Evol. Equ. 14, 333–356 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Becker, A. Jentzen, P.E. Kloeden, An exponential Wagner-Platen type scheme for SPDEs. SIAM J. Numer. Anal. 54, 2389–2426 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Ben Arous, M. Grădinaru, M. Ledoux, Hölder norms and the support theorem for diffusions. Ann. Inst. H. Poincaré Probab. Stat. 30, 415–436 (1994)

    MATH  Google Scholar 

  18. A. Bensoussan, R. Glowinski, A. Răşcanu, Approximation of the Zakai equation by the splitting up method. SIAM J. Control Optim. 28, 1420–1431 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Bensoussan, R. Glowinski, A. Răşcanu, Approximation of some stochastic differential equations by the splitting up method. Appl. Math. Optim. 25, 81–106 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. F.E. Benth, J. Gjerde, Convergence rates for finite element approximations of stochastic partial differential equations. Stoch. Stoch. Rep. 63, 313–326 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Blömker, A. Jentzen, Galerkin approximations for the stochastic Burgers equation. SIAM J. Numer. Anal. 51, 694–715 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Blömker, M. Kamrani, S.M. Hosseini, Full discretization of the stochastic Burgers equation with correlated noise. IMA J. Numer. Anal. 33(3), 825–848 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Breckner, Approximation of the solution of the stochastic Navier-Stokes equation. Optimization 49, 15–38 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Z. Brzeźniak, E. Carelli, A. Prohl, Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing. IMA J. Numer. Anal. 33, 771–824 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Z. Brzeźniak, F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations. Stoch. Process. Appl. 55, 329–358 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Z. Brzeźniak, A. Millet, On the splitting method for some complex-valued quasilinear evolution equations, in Stochastic Analysis and Related Topics, ed. by L. Decreusefond, J. Najim (Springer, Berlin, 2012), pp. 57–90

    Chapter  Google Scholar 

  27. E. Buckwar, T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. Math. Comput. Simul. 81, 1110–1127 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Budhiraja, G. Kallianpur, Approximations to the solution of the Zakai equation using multiple Wiener and Stratonovich integral expansions. Stoch. Stoch. Rep. 56, 271–315 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Budhiraja, G. Kallianpur, The Feynman-Stratonovich semigroup and Stratonovich integral expansions in nonlinear filtering. Appl. Math. Optim. 35, 91–116 (1997)

    MathSciNet  MATH  Google Scholar 

  30. Y. Cao, H. Yang, L. Yin, Finite element methods for semilinear elliptic stochastic partial differential equations. Numer. Math. 106, 181–198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Y. Cao, L. Yin, Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Commun. Pure Appl. Anal. 6, 607–617 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. E. Carelli, A. Prohl, Rates of convergence for discretizations of the stochastic incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 50, 2467–2496 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. H.D. Ceniceros, G.O. Mohler, A practical splitting method for stiff SDEs with applications to problems with small noise. Multiscale Model. Simul. 6, 212–227 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Chaleyat-Maurel, D. Michel, A Stroock Varadhan support theorem in nonlinear filtering theory. Probab. Theory Relat. Fields 84, 119–139 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50, 216–246 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Charrier, A. Debussche, Weak truncation error estimates for elliptic PDEs with lognormal coefficients. Stoch. PDE: Anal. Comp. 1, 63–93 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. G.-Q. Chen, Q. Ding, K.H. Karlsen, On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204, 707–743 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. P. L. Chow, J.-L. Jiang, J.-L. Menaldi, Pathwise convergence of approximate solutions to Zakai’s equation in a bounded domain, in Stochastic Partial Differential Equations and Applications, Trento, 1990 (Longman Scientific & Technical, Harlow, 1992), pp. 111–123

    MATH  Google Scholar 

  39. I. Chueshov, A. Millet, Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem. Stoch. Anal. Appl. 29, 570–611 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. S. Cox, J. van Neerven, Convergence rates of the splitting scheme for parabolic linear stochastic Cauchy problems. SIAM J. Numer. Anal. 48, 428–451 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. S. Cox, J. van Neerven, Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise. Numer. Math. 125, 259–345 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. D. Crisan, Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation. Ann. Probab. 31, 693–718 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. D. Crisan, J. Xiong, Numerical solutions for a class of SPDEs over bounded domains, in Conference Oxford sur les méthodes de Monte Carlo séquentielles (EDP Sciences, Les Ulis, 2007), pp. 121–125

    MATH  Google Scholar 

  44. G. Da Prato, Kolmogorov Equations for Stochastic PDEs (Birkhäuser, Basel, 2004)

    Book  MATH  Google Scholar 

  45. G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions (Cambridge University Press, Cambridge, 1992)

    Book  MATH  Google Scholar 

  46. A.M. Davie, J.G. Gaines, Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. Math. Comp. 70, 121–134 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  47. A. de Bouard, A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise. Commun. Math. Phys. 205, 161–181 (1999)

    Article  MATH  Google Scholar 

  48. A. de Bouard, A. Debussche, Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation. Appl. Math. Optim. 54, 369–399 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  49. A. de Bouard, A. Debussche, Random modulation of solitons for the stochastic Korteweg-de Vries equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 251–278 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. A. de Bouard, A. Debussche, The nonlinear Schrödinger equation with white noise dispersion. J. Funct. Anal. 259, 1300–1321 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. A. de Bouard, A. Debussche, L. Di Menza, Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations. Monte Carlo Methods Appl. 7, 55–63 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  52. A. de Bouard, A. Debussche, Y. Tsutsumi, White noise driven Korteweg-de Vries equation. J. Funct. Anal. 169, 532–558 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  53. A. de Bouard, A. Debussche, On the stochastic Korteweg-de Vries equation. J. Funct. Anal. 154, 215–251 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  54. A. Debussche, The 2D-Navier-Stokes equations perturbed by a delta correlated noise, in Probabilistic Methods in Fluids (World Scientific Publishers, River Edge, NJ, 2003), pp. 115–129

    MATH  Google Scholar 

  55. A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case. Math. Comp. 80, 89–117 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  56. A. Debussche, J. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation. Phys. D 134, 200–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  57. A. Debussche, J. Printems, Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete Contin. Dyn. Syst. Ser. B 6, 761–781 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  58. A. Debussche, J. Printems, Weak order for the discretization of the stochastic heat equation. Math. Comp. 78, 845–863 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. A. Debussche, J. Vovelle, Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259, 1014–1042 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  60. Q. Du, T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises. SIAM J. Numer. Anal. 40, 1421–1445 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  61. Y. Duan, X. Yang, On the convergence of a full discretization scheme for the stochastic Navier-Stokes equations. J. Comput. Anal. Appl. 13, 485–498 (2011)

    MathSciNet  MATH  Google Scholar 

  62. Y. Duan, X. Yang, The finite element method of a Euler scheme for stochastic Navier-Stokes equations involving the turbulent component. Int. J. Numer. Anal. Model. 10, 727–744 (2013)

    MathSciNet  MATH  Google Scholar 

  63. M.A. El-Tawil, A.-H.A. El-Shikhipy, Approximations for some statistical moments of the solution process of stochastic Navier-Stokes equation using WHEP technique. Appl. Math. Inf. Sci. 6, 1095–1100 (2012)

    MathSciNet  Google Scholar 

  64. J. Feng, D. Nualart, Stochastic scalar conservation laws. J. Funct. Anal. 255, 313–373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  65. F. Flandoli, V.M. Tortorelli, Time discretization of Ornstein-Uhlenbeck equations and stochastic Navier-Stokes equations with a generalized noise. Stoch. Stoch. Rep. 55, 141–165 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  66. P. Florchinger, F. Le Gland, Time-discretization of the Zakai equation for diffusion processes observed in correlated noise. Stoch. Stoch. Rep. 35, 233–256 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  67. P. Friz, H. Oberhauser, On the splitting-up method for rough (partial) differential equations. J. Differ. Equ. 251, 316–338 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  68. J.G. Gaines, Numerical experiments with S(P)DE’s, in Stochastic Partial Differential Equations (Edinburgh, 1994) (Cambridge University Press, Cambridge, 1995), pp. 55–71

    Book  MATH  Google Scholar 

  69. J. Galvis, M. Sarkis, Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. SIAM J. Numer. Anal. 47, 3624–3651 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Ganguly, Wong-Zakai type convergence in infinite dimensions. Electron. J. Probab. 18(31), 34 (2013)

    Google Scholar 

  71. M. Geissert, M. Kovács, S. Larsson, Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise. BIT Numer. Math. 49, 343–356 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  72. M. Gerencsér, I. Gyöngy, Finite difference schemes for stochastic partial differential equations in Sobolev spaces. Appl. Math. Optim. 72, 77–100 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  73. A. Germani, M. Piccioni, Semidiscretization of stochastic partial differential equations on R d by a finite-element technique. Stochastics 23, 131–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  74. M.B. Giles, C. Reisinger, Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIAM J. Financ. Math. 3, 572–592 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  75. N.Y. Goncharuk, P. Kotelenez, Fractional step method for stochastic evolution equations. Stoch. Process. Appl. 73, 1–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  76. W. Grecksch, P.E. Kloeden, Time-discretised Galerkin approximations of parabolic stochastic PDEs. Bull. Aust. Math. Soc. 54, 79–85 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  77. W. Grecksch, H. Lisei, Approximation of stochastic nonlinear equations of Schrödinger type by the splitting method. Stoch. Anal. Appl. 31, 314–335 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  78. W. Grecksch, B. Schmalfuß, Approximation of the stochastic Navier-Stokes equation. Mat. Apl. Comput. 15, 227–239 (1996)

    MathSciNet  MATH  Google Scholar 

  79. C. Gugg, H. Kielhöfer, M. Niggemann, On the approximation of the stochastic Burgers equation. Commun. Math. Phys. 230, 181–199 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  80. B. Gustafsson, H.-O. Kreiss, J. Oliger, Time Dependent Problems and Difference Methods (Wiley, New York, 1995)

    MATH  Google Scholar 

  81. I. Gyöngy, On the approximation of stochastic partial differential equations. I. Stochastics 25, 59–85 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  82. I. Gyöngy, On the approximation of stochastic partial differential equations. II. Stoch. Stoch. Rep. 26, 129–164 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  83. I. Gyöngy, The stability of stochastic partial differential equations and applications. I. Stoch. Stoch. Rep. 27, 129–150 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  84. I. Gyöngy, On the support of the solutions of stochastic differential equations. Teor. Veroyatnost. i Primenen. 39, 649–653 (1994)

    MathSciNet  MATH  Google Scholar 

  85. I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I. Potential Anal. 9, 1–25 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  86. I. Gyöngy, Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. II. Potential Anal. 11, 1–37 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  87. I. Gyöngy, Approximations of stochastic partial differential equations, in Stochastic Partial Differential Equations and Applications (Trento, 2002) (Dekker, New York, 2002), pp. 287–307

    MATH  Google Scholar 

  88. I. Gyöngy, N. Krylov, On the rate of convergence of splitting-up approximations for SPDEs, in Stochastic Inequalities and Applications (Birkhäuser, Basel, 2003), pp. 301–321

    Book  MATH  Google Scholar 

  89. I. Gyöngy, N. Krylov, On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31, 564–591 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  90. I. Gyöngy, N. Krylov, An accelerated splitting-up method for parabolic equations. SIAM J. Math. Anal. 37, 1070–1097 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  91. I. Gyöngy, N. Krylov, Accelerated finite difference schemes for linear stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 42, 2275–2296 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  92. I. Gyöngy, T. Martínez, On numerical solution of stochastic partial differential equations of elliptic type. Stochastics 78, 213–231 (2006)

    MathSciNet  MATH  Google Scholar 

  93. I. Gyöngy, D. Nualart, Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise. Stoch. Process. Appl. 58, 57–72 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  94. I. Gyöngy, D. Nualart, Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Anal. 7, 725–757 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  95. I. Gyöngy, T. Pröhle, On the approximation of stochastic differential equation and on Stroock-Varadhan’s support theorem. Comput. Math. Appl. 19, 65–70 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  96. I. Gyöngy, A. Shmatkov, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54, 315–341 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  97. I. Gyöngy, P.R. Stinga, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, in Seminar on Stochastic Analysis, Random Fields and Applications VII, ed. by R.C. Dalang, M. Dozzi, F. Russo (Springer, Basel, 2013), pp. 95–130

    Chapter  Google Scholar 

  98. M. Hairer, J. Maas, A spatial version of the Itô-Stratonovich correction. Ann. Probab. 40, 1675–1714 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  99. M. Hairer, J. Voss, Approximations to the stochastic Burgers equation. J. Nonlinear Sci. 21, 897–920 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  100. E.J. Hall, Accelerated spatial approximations for time discretized stochastic partial differential equations. SIAM J. Math. Anal. 44, 3162–3185 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  101. E.J. Hall, Higher order spatial approximations for degenerate parabolic stochastic partial differential equations. SIAM J. Math. Anal. 45, 2071–2098 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  102. E. Hausenblas, Numerical analysis of semilinear stochastic evolution equations in Banach spaces. J. Comput. Appl. Math. 147, 485–516 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  103. E. Hausenblas, Approximation for semilinear stochastic evolution equations. Potential Anal. 18, 141–186 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  104. E. Hausenblas, Weak approximation for semilinear stochastic evolution equations, in Stochastic Analysis and Related Topics VIII (Birkhäuser, Basel, 2003), pp. 111–128

    MATH  Google Scholar 

  105. E. Hausenblas, Weak approximation of the stochastic wave equation. J. Comput. Appl. Math. 235, 33–58 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  106. J. He, Numerical analysis for stochastic age-dependent population equations with diffusion, in Advances in Electronic Commerce, Web Application and Communication, ed. by D. Jin, S. Lin (Springer, Berlin, 2012), pp. 37–43

    Chapter  Google Scholar 

  107. R.L. Herman, A. Rose, Numerical realizations of solutions of the stochastic KdV equation. Math. Comput. Simul. 80, 164–172 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  108. J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems. Cambridge Monographs on Applied and Computational Mathematics, vol. 21 (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  109. D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  110. D.J. Higham, X. Mao, A.M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  111. D.J. Higham, X. Mao, L. Szpruch, Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete Contin. Dyn. Syst. Ser. B 18, 2083–2100 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  112. H. Holden, N.H. Risebro, Conservation laws with a random source. Appl. Math. Optim. 36, 229–241 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  113. Y. Hu, G. Kallianpur, J. Xiong, An approximation for the Zakai equation. Appl. Math. Optim. 45, 23–44 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  114. M. Hutzenthaler, A. Jentzen, On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients. ArXiv (2014). https://arxiv.org/abs/1401.0295

  115. M. Hutzenthaler, A. Jentzen, Numerical approximation of stochastic differential equations with non-globally Lipschitz continuous coefficients. Mem. Am. Math. Soc. 236, 1 (2015)

    MathSciNet  MATH  Google Scholar 

  116. M. Hutzenthaler, A. Jentzen, P.E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22, 1611–1641 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  117. M. Hutzenthaler, A. Jentzen, X. Wang, Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations. Math. Comp. (2017). https://doi.org/10.1090/mcom/3146

    Google Scholar 

  118. S.M. Iacus, Simulation and Inference for Stochastic Differential Equations. Springer Series in Statistics (Springer, New York, 2008). With R examples

    Google Scholar 

  119. K. Ito, Approximation of the Zakai equation for nonlinear filtering. SIAM J. Control Optim. 34, 620–634 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  120. K. Ito, B. Rozovskii, Approximation of the Kushner equation for nonlinear filtering. SIAM J. Control Optim. 38, 893–915 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  121. A. Jentzen, Pathwise numerical approximation of SPDEs with additive noise under non-global Lipschitz coefficients. Potential Anal. 31, 375–404 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  122. A. Jentzen, Higher order pathwise numerical approximations of SPDEs with additive noise. SIAM J. Numer. Anal. 49, 642–667 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  123. A. Jentzen, P.E. Kloeden, The numerical approximation of stochastic partial differential equations. Milan J. Math. 77, 205–244 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  124. A. Jentzen, 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, 649–667 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  125. A. Jentzen, M. Röckner, A Milstein scheme for SPDEs. Found. Comput. Math. 15, 313–362 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  126. I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, New York, 1991)

    MATH  Google Scholar 

  127. P.E. Kloeden, A. Jentzen, Pathwise convergent higher order numerical schemes for random ordinary differential equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463, 2929–2944 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  128. P.E. Kloeden, G.J. Lord, A. Neuenkirch, T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds. J. Comput. Appl. Math. 235, 1245–1260 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  129. P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992)

    Book  MATH  Google Scholar 

  130. P.E. Kloeden, S. Shott, Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs. J. Appl. Math. Stoch. Anal. 14, 47–53 (2001)

    Article  MATH  Google Scholar 

  131. G.T. Kossioris, G.E. Zouraris, Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise. M2AN Math. Model. Numer. Anal. 44, 289–322 (2010)

    Google Scholar 

  132. G.T. Kossioris, G.E. Zouraris, Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete Contin. Dyn. Syst. Ser. B 18, 1845–1872 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  133. G.T. Kossioris, G.E. Zouraris, Finite element approximations for a linear fourth-order parabolic SPDE in two and three space dimensions with additive space-time white noise. Appl. Numer. Math. 67, 243–261 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  134. M. Kovács, S. Larsson, F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise. BIT Numer. Math. 52, 85–108 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  135. M. Kovács, S. Larsson, F. Lindgren, Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes. BIT Numer. Math. 53, 497–525 (2013)

    MATH  Google Scholar 

  136. M. Kovács, S. Larsson, A. Mesforush, Finite element approximation of the Cahn-Hilliard-Cook equation. SIAM J. Numer. Anal. 49, 2407–2429 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  137. M. Kovács, S. Larsson, F. Saedpanah, Finite element approximation of the linear stochastic wave equation with additive noise. SIAM J. Numer. Anal. 48, 408–427 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  138. I. Kröker, C. Rohde, Finite volume schemes for hyperbolic balance laws with multiplicative noise. Appl. Numer. Math. 62, 441–456 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  139. R. Kruse, Consistency and stability of a Milstein-Galerkin finite element scheme for semilinear SPDE. Stoch. Partial Differ. Equ. Anal. Comput. 2, 471–516 (2014)

    MathSciNet  MATH  Google Scholar 

  140. R. Kruse, Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise. IMA J. Numer. Anal. 34, 217–251 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  141. T.G. Kurtz, J. Xiong, Numerical solutions for a class of SPDEs with application to filtering, in Stochastics in Finite and Infinite Dimensions. Trends in Mathematics (Birkhäuser, Boston, MA, 2001), pp. 233–258

    Google Scholar 

  142. H.J. Kushner, On the differential equations satisfied by conditional probability densities of Markov processes, with applications. J. Soc. Indust. Appl. Math. Ser. A Control 2, 106–119 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  143. A. Lang, A Lax equivalence theorem for stochastic differential equations. J. Comput. Appl. Math. 234, 3387–3396 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  144. A. Lang, Almost sure convergence of a Galerkin approximation for SPDEs of Zakai type driven by square integrable martingales. J. Comput. Appl. Math. 236, 1724–1732 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  145. A. Lang, P.-L. Chow, J. Potthoff, Almost sure convergence for a semidiscrete Milstein scheme for SPDEs of Zakai type. Stochastics 82, 315–326 (2010)

    MathSciNet  MATH  Google Scholar 

  146. S. Larsson, A. Mesforush, Finite-element approximation of the linearized Cahn-Hilliard-Cook equation. IMA J. Numer. Anal. 31, 1315–1333 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  147. F. Le Gland, Splitting-up approximation for SPDEs and SDEs with application to nonlinear filtering, in Stochastic Partial Differential Equations and Their Applications (Springer, Berlin, 1992), pp. 177–187

    MATH  Google Scholar 

  148. R.J. LeVeque, Numerical Methods for Conservation Laws. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 1990)

    Google Scholar 

  149. F. Lindner, R. Schilling, Weak order for the discretization of the stochastic heat equation driven by impulsive noise. Potential Anal. 38, 345–379 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  150. D. Liu, Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise. Commun. Math. Sci. 1, 361–375 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  151. J. Liu, A mass-preserving splitting scheme for the stochastic Schrödinger equation with multiplicative noise. IMA J. Numer. Anal. 33, 1469–1479 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  152. J. Liu, Order of convergence of splitting schemes for both deterministic and stochastic nonlinear Schrödinger equations. SIAM J. Numer. Anal. 51, 1911–1932 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  153. J.A. Londoño, A.M. Ramirez, Numerical performance of some Wong-Zakai type approximations for stochastic differential equations, Technical report, Department of Mathematics, National University of Colombia, Bogota, Colombia, 2006

    Google Scholar 

  154. G.J. Lord, J. Rougemont, A numerical scheme for stochastic PDEs with Gevrey regularity. IMA J. Numer. Anal. 24, 587–604 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  155. G.J. Lord, T. Shardlow, Postprocessing for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 45, 870–889 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  156. G.J. Lord, A. Tambue, A modified semi-implicit Euler-Maruyama scheme for finite element discretization of SPDEs. ArXiv (2010). https://arxiv.org/abs/1004.1998

  157. G.J. Lord, A. Tambue, Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise. IMA J. Numer. Anal. 33, 515–543 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  158. G.J. Lord, V. Thümmler, Computing stochastic traveling waves. SIAM J. Sci. Comput. 34, B24–B43 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  159. S. Lototsky, B. Rozovskii, Stochastic differential equations: a Wiener chaos approach, in From Stochastic Calculus to Mathematical Finance (Springer, Berlin, 2006), pp. 433–506

    Book  Google Scholar 

  160. S.V. Lototsky, B.L. Rozovskii, Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab. 34, 638–662 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  161. R. Marty, On a splitting scheme for the nonlinear Schrödinger equation in a random medium. Commun. Math. Sci. 4, 679–705 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  162. R. Mikulevicius, B. Rozovskii, Linear parabolic stochastic PDEs and Wiener chaos. SIAM J. Math. Anal. 29, 452–480 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  163. R. Mikulevicius, B.L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35, 1250–1310 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  164. A. Millet, P.-L. Morien, On implicit and explicit discretization schemes for parabolic SPDEs in any dimension. Stoch. Process. Appl. 115, 1073–1106 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  165. A. Millet, M. Sanz-Solé, A simple proof of the support theorem for diffusion processes, in Séminaire de Probabilités, XXVIII. Lecture Notes in Mathematics, vol. 1583 (Springer, Berlin, 1994), pp. 36–48

    Google Scholar 

  166. G.N. Mil′shteĭn, A theorem on the order of convergence of mean-square approximations of solutions of systems of stochastic differential equations. Teor. Veroyatnost. i Primenen. 32, 809–811 (1987)

    Google Scholar 

  167. G.N. Milstein, Numerical Integration of Stochastic Differential Equations (Kluwer Academic Publishers Group, Dordrecht, 1995)

    Book  MATH  Google Scholar 

  168. G.N. Milstein, M.V. Tretyakov, Stochastic Numerics for Mathematical Physics (Springer, Berlin, 2004)

    Book  MATH  Google Scholar 

  169. G.N. Milstein, M.V. Tretyakov, Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. SIAM J. Numer. Anal. 43, 1139–1154 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  170. G.N. Milstein, M.V. Tretyakov, Solving parabolic stochastic partial differential equations via averaging over characteristics. Math. Comp. 78, 2075–2106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  171. J. Ming, M. Gunzburger, Efficient numerical methods for stochastic partial differential equations through transformation to equations driven by correlated noise. Int. J. Uncertain. Quant. 3, 321–339 (2013)

    Article  MathSciNet  Google Scholar 

  172. Y.S. Mishura, G.M. Shevchenko, Approximation schemes for stochastic differential equations in a Hilbert space. Teor. Veroyatn. Primen. 51, 476–495 (2006)

    Article  MathSciNet  Google Scholar 

  173. K. Mohamed, M. Seaid, M. Zahri, A finite volume method for scalar conservation laws with stochastic time-space dependent flux functions. J. Comput. Appl. Math. 237, 614–632 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  174. C.M. Mora, Numerical solution of conservative finite-dimensional stochastic Schrödinger equations. Ann. Appl. Probab. 15, 2144–2171 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  175. T. Müller-Gronbach, K. Ritter, An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise. BIT Numer. Math. 47, 393–418 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  176. T. Müller-Gronbach, K. Ritter, Lower bounds and nonuniform time discretization for approximation of stochastic heat equations. Found. Comput. Math. 7, 135–181 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  177. J. Picard, Approximation of nonlinear filtering problems and order of convergence, in Filtering and Control of Random Processes (Paris, 1983) (Springer, Berlin, 1984), pp. 219–236

    Google Scholar 

  178. J. Printems, On the discretization in time of parabolic stochastic partial differential equations. M2AN Math. Model. Numer. Anal. 35, 1055–1078 (2001)

    Google Scholar 

  179. R. Qi, X. Yang, Weak convergence of finite element method for stochastic elastic equation driven by additive noise. J. Sci. Comput. 56, 450–470 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  180. L. Quer-Sardanyons, M. Sanz-Solé, Space semi-discretisations for a stochastic wave equation. Potential Anal. 24, 303–332 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  181. C. Reisinger, Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative. Int. J. Comput. Math. 89, 2562–2575 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  182. A.J. Roberts, A step towards holistic discretisation of stochastic partial differential equations. ANZIAM J. 45, C1–C15 (2003/04)

    Google Scholar 

  183. B.L. Rozovskiĭ, Stochastic Evolution Systems (Kluwer, Dordecht, 1990)

    Book  Google Scholar 

  184. M. Sango, Splitting-up scheme for nonlinear stochastic hyperbolic equations. Forum Math. 25, 931–965 (2013)

    MathSciNet  MATH  Google Scholar 

  185. B. Saussereau, I.L. Stoica, Scalar conservation laws with fractional stochastic forcing: existence, uniqueness and invariant measure. Stoch. Process. Appl. 122, 1456–1486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  186. B. Schmalfuss, On approximation of the stochastic Navier-Stokes equations. Wiss. Z. Tech. Hochsch. Leuna-Merseburg 27, 605–612 (1985)

    MathSciNet  MATH  Google Scholar 

  187. H. Schurz, Numerical analysis of stochastic differential equations without tears, in Handbook of Stochastic Analysis and Applications (Dekker, New York, 2002), pp. 237–359

    MATH  Google Scholar 

  188. T. Shardlow, Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim. 20, 121–145 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  189. T. Shardlow, Weak convergence of a numerical method for a stochastic heat equation. BIT Numer. Math. 43, 179–193 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  190. G. Shevchenko, Rate of convergence of discrete approximations of solutions to stochastic differential equations in a Hilbert space. Theory Probab. Math. Stat. 69, 187–199 (2004)

    Article  MathSciNet  Google Scholar 

  191. A.R. Soheili, M.B. Niasar, M. Arezoomandan, Approximation of stochastic parabolic differential equations with two different finite difference schemes. Bull. Iran. Math. Soc. 37, 61–83 (2011)

    MathSciNet  MATH  Google Scholar 

  192. V.N. Stanciulescu, M.V. Tretyakov, Numerical solution of the Dirichlet problem for linear parabolic SPDEs based on averaging over characteristics, in Stochastic Analysis 2010 (Springer, Heidelberg, 2011), pp. 191–212

    Google Scholar 

  193. M. Sun, R. Glowinski, Pathwise approximation and simulation for the Zakai filtering equation through operator splitting. Calcolo 30, 219–239 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  194. G. Tessitore, J. Zabczyk, Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6, 621–655 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  195. T.G. Theting, Numerical solution of Wick-stochastic partial differential equations, in Proceedings of the International Conference on Stochastic Analysis and Applications (Kluwer Academic Publishers, Dordrecht, 2004), pp. 303–349

    MATH  Google Scholar 

  196. M.V. Tretyakov, Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications. SIAM J. Numer. Anal. 51, 3135–3162 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  197. K. Twardowska, An extension of the Wong-Zakai theorem for stochastic evolution equations in Hilbert spaces. Stoch. Anal. Appl. 10, 471–500 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  198. K. Twardowska, An approximation theorem of Wong-Zakai type for nonlinear stochastic partial differential equations. Stoch. Anal. Appl. 13, 601–626 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  199. K. Twardowska, An approximation theorem of Wong-Zakai type for stochastic Navier-Stokes equations. Rend. Sem. Mat. Univ. Padova 96, 15–36 (1996)

    MathSciNet  MATH  Google Scholar 

  200. K. Twardowska, Wong-Zakai approximations for stochastic differential equations. Acta Appl. Math. 43, 317–359 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  201. J.B. Walsh, Finite element methods for parabolic stochastic PDE’s. Potential Anal. 23, 1–43 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  202. J.B. Walsh, On numerical solutions of the stochastic wave equation. Ill. J. Math. 50, 991–1018 (2006)

    MathSciNet  MATH  Google Scholar 

  203. X. Wan, B. Rozovskii, G.E. Karniadakis, A stochastic modeling methodology based on weighted Wiener chaos and Malliavin calculus. Proc. Natl. Acad. Sci. U.S.A. 106, 14189–14194 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  204. X. Wang, S. Gan, A Runge-Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise. Numer. Algorithms 62, 193–223 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  205. X. Wang, S. Gan, Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise. J. Math. Anal. Appl. 398, 151–169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  206. X. Wang, S. Gan, J. Tang, Higher order strong approximations of semilinear stochastic wave equation with additive space-time white noise. SIAM J. Sci. Comput. 36, A2611–A2632 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  207. E. Wong, M. Zakai, On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36, 1560–1564 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  208. Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43, 1363–1384 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  209. X. Yang, Y. Duan, Y. Guo, A posteriori error estimates for finite element approximation of unsteady incompressible stochastic Navier-Stokes equations. SIAM J. Numer. Anal. 48, 1579–1600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  210. X. Yang, W. Wang, Y. Duan, The approximation of a Crank-Nicolson scheme for the stochastic Navier-Stokes equations. J. Comput. Appl. Math. 225, 31–43 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  211. R.-M. Yao, L.-J. Bo, Discontinuous Galerkin method for elliptic stochastic partial differential equations on two and three dimensional spaces. Sci. China Ser. A 50, 1661–1672 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  212. H. Yoo, Semi-discretization of stochastic partial differential equations on R 1 by a finite-difference method. Math. Comp. 69, 653–666 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  213. N. Yoshida, Stochastic shear thickening fluids: strong convergence of the Galerkin approximation and the energy equality. Ann. Appl. Probab. 22, 1215–1242 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  214. L. Zhang, Q.M. Zhang, Convergence of numerical solutions for the stochastic Navier-Stokes equation. Math. Appl. (Wuhan) 21, 504–509 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts, USA

    Zhongqiang Zhang

  2. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    George Em Karniadakis

Authors
  1. Zhongqiang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. George Em Karniadakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Zhang, Z., Karniadakis, G.E. (2017). Numerical methods for stochastic differential equations. In: Numerical Methods for Stochastic Partial Differential Equations with White Noise. Applied Mathematical Sciences, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-319-57511-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation