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SPARSE GRID STOCHASTIC COLLOCATION METHOD FOR STOCHASTIC BURGERS EQUATION

  • Lee, Hyung-Chun (Department of Mathematics Ajou University) ;
  • Nam, Yun (Department of Mathematics Ajou University)
  • Received : 2015.11.08
  • Published : 2017.01.01

Abstract

We investigate an efficient approximation of solution to stochastic Burgers equation driven by an additive space-time noise. We discuss existence and uniqueness of a solution through the Orstein-Uhlenbeck (OU) process. To approximate the OU process, we introduce the Karhunen-$Lo{\grave{e}}ve$ expansion, and sparse grid stochastic collocation method. About spatial discretization of Burgers equation, two separate finite element approximations are presented: the conventional Galerkin method and Galerkin-conservation method. Numerical experiments are provided to demonstrate the efficacy of schemes mentioned above.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. E. Allen, S. Novosel, and Z. Zhang, Finite element and difference approximation of some linear stochastic partial differential equations, Stoch. Stoch. Rep. 64 (1998), no. 1-2, 117-142. https://doi.org/10.1080/17442509808834159
  2. K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, second ed., Springer, New York, 2005.
  3. I. Babuska, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45 (2007), no. 3, 1005-1034. https://doi.org/10.1137/050645142
  4. V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (2000), no. 4, 273-288. https://doi.org/10.1023/A:1018977404843
  5. D. Blomker and A. Jentzen, Garlerkin approximations for the stochastic Burgers equation, SIAM J. Numer. Anal. 51 (2013), no. 1, 694-715. https://doi.org/10.1137/110845756
  6. H. J. Bungartz and M. Griebel, Sparse grids, Acta Numer. 13 (2004), 147-269. https://doi.org/10.1017/S0962492904000182
  7. A. J. Chorin and O. Hald, Stochastic Tools in Mathematics and Science, second ed., Springer, New York, 2009.
  8. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 2, Wiley-Interscience, New York, 1953.
  9. G. Da Prato, A. Debussche, and R. Temam, Stochastic Burgers' equation, Nonlinear Differential Equations Appl. 1 (1994), no. 4, 389-402. https://doi.org/10.1007/BF01194987
  10. G. Da Prato and D. Gatarek, Stochastic Burgers equation with correlated noise, Stoch. Stoch. Rep. 52 (1995), no. 1-2, 29-41. https://doi.org/10.1080/17442509508833962
  11. G. Da Prato and J. Zabcayk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 2008.
  12. Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by spectral additive noises, SIAM J. Numer. Anal. 40 (2003), 1421-1445.
  13. L. C. Evans, Partial Differential Equations, second ed. American Mathematical Society, Providence, RI, 2010.
  14. C. A. J. Fletcher, Computational Garlerkin methods, Springer-Verlag, New York, 1984.
  15. R. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, revised ed., Dover Publications, New York, 2003.
  16. 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), no. 2102, 649-667. https://doi.org/10.1098/rspa.2008.0325
  17. M. Loeve, Probability Theory, fourth ed., Springer, New York, 1977.
  18. J. Ming and M. Gunzburger, Efficient numerical method for stochastic partial differential equations through transformation to equations driven by correlated noise, Int. J. Uncertain. Quantif. 3 (2013), no. 4, 321-329. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2012003670
  19. E. Novak and K. Ritter, High dimensional integration of smooth functions over cubes, Numer. Math. 75 (1996), no. 1, 79-97. https://doi.org/10.1007/s002110050231
  20. E. Novak, K. Ritter, R. Schmitt, and A. Steinbauer, On an interpolatory method for high dimensional integration, J. Comput. Appl. Math. 112 (1999), no. 1-2, 215-228. https://doi.org/10.1016/S0377-0427(99)00222-8
  21. F. Nobile, R. Tempone, and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal. 46 (2008), no. 5, 2309-2345. https://doi.org/10.1137/060663660
  22. F. Nobile, R. Tempone, and C. G. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal. 46 (2008), no. 5, 2411-2442. https://doi.org/10.1137/070680540
  23. M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, second ed., Springer, New York, 2004.
  24. S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 4 (1963), 240-243.
  25. G. W. Wasilkowski and H. Wozniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11 (1995), no. 1, 1-56. https://doi.org/10.1006/jcom.1995.1001