On the discretization in time of parabolic stochastic partial differential equations

Jacques Printems

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 6, page 1055-1078
  • ISSN: 0764-583X

Abstract

top
We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case.

How to cite

top

Printems, Jacques. "On the discretization in time of parabolic stochastic partial differential equations." ESAIM: Mathematical Modelling and Numerical Analysis 35.6 (2010): 1055-1078. <http://eudml.org/doc/197506>.

@article{Printems2010,
abstract = { We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case. },
author = {Printems, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Stochastic partial differential equations; semi-discretized scheme for stochastic partial differential equations; Euler scheme.; stochastic partial differential equation; semi-discretization; Euler scheme},
language = {eng},
month = {3},
number = {6},
pages = {1055-1078},
publisher = {EDP Sciences},
title = {On the discretization in time of parabolic stochastic partial differential equations},
url = {http://eudml.org/doc/197506},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Printems, Jacques
TI - On the discretization in time of parabolic stochastic partial differential equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 6
SP - 1055
EP - 1078
AB - We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case.
LA - eng
KW - Stochastic partial differential equations; semi-discretized scheme for stochastic partial differential equations; Euler scheme.; stochastic partial differential equation; semi-discretization; Euler scheme
UR - http://eudml.org/doc/197506
ER -

References

top
  1. A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stoke. J. Funct. Anal.13 (1973) 195-222.  
  2. J.H. Bramble, A.H. Schatz, V. Thomée and L.B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal.14 (1977) 218-241.  
  3. C. Cardon-Weber, Autour d'équations aux dérivées partielles stochastiques à dérives non-Lipschitziennes. Thèse, Université Paris VI, Paris (2000).  
  4. M. Crouzeix and V. Thomée, On the discretization in time of semilinear parabolic equations with nonsmooth initial data. Math. Comput.49 (1987) 359-377.  
  5. G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation. Nonlinear Anal., Theory Methods. Appl.26 (1996) 241-263.  
  6. G. Da Prato, A. Debussche and R. Temam, Stochastic Burgers' equation. Nonlinear Differ. Equ. Appl.1 (1994) 389-402.  
  7. G. Da Prato and D. Gatarek, Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep.52 (1995) 29-41.  
  8. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, in Encyclopedia of Mathematics and its Application. Cambridge University Press, Cambridge (1992).  
  9. F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Relat. Fields102 (1995) 367-391.  
  10. 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.  
  11. 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.  
  12. 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.  
  13. I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process Appl.73 (1998) 271-299.  
  14. C. Johnson, S. Larsson, V. Thomée and L.B. Wahlbin, Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data. Math. Comput.49 (1987) 331-357.  
  15. P.E. Kloeden and E. Platten, Numerical solution of stochastic differential equations, in Applications of Mathematics 23, Springer-Verlag, Berlin, Heidelberg, New York (1992).  
  16. N. Krylov and B.L. Rozovski, Stochastic Evolution equations. J. Sov. Math.16 (1981) 1233-1277.  
  17. M.-N. Le Roux, Semidiscretization in Time for Parabolic Problems. Math. Comput.33 (1979) 919-931.  
  18. G.N. Milstein, Approximate integration of stochastic differential equations. Theor. Prob. Appl.19 (1974) 557-562.  
  19. E. Pardoux, Équations aux dérivées partielles stochastiques non linéaires monotones. Étude de solutions fortes de type Ito. Thèse, Université Paris XI, Orsay (1975).  
  20. B.L. Rozozski, Stochastic evolution equations. Linear theory and application to nonlinear filtering. Kluwer, Dordrecht, The Netherlands (1990).  
  21. T. Shardlow, Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optimization20 (1999) 121-145.  
  22. D. Talay, Efficient numerical schemes for the approximation of expectation of functionals of the solutions of an stochastic differential equation and applications, in Lecture Notes in Control and Information Science 61, Springer, London, (1984) 294-313.  
  23. D. Talay, Discrétisation d'une équation différentielle stochastique et calcul approché d'espérance de fonctionnelles de la solution. RAIRO Modél. Math. Anal. Numér.20 (1986) 141-179.  
  24. M. Viot, Solutions faibles aux équations aux dérivées partielles stochastiques non linéaires. Thèse, Université Pierre et Marie Curie, Paris (1976).  
  25. J. B. Walsh, An introduction to stochastic partial differential equations, in Lectures Notes in Mathematics 1180 (1986) 265-437.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.