An efficient computational method for statistical moments of Burger's equation with random initial conditions

Kim, Hongjoong

doi:https://doi.org/10.1155/MPE/2006/17406

Mathematical Problems in Engineering

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 017406 | https://doi.org/10.1155/MPE/2006/17406

An efficient computational method for statistical moments of Burger's equation with random initial conditions

Hongjoong Kim¹

Received15 Nov 2005

Revised04 Jul 2006

Accepted12 Sept 2006

Published16 Nov 2006

Abstract

The paper is concerned with efficient computation of numerical solutions to Burger's equation with random initial conditions. When the Lax-Wendroff scheme (LW) is expanded using the Wiener chaos expansion (WCE), random and deterministic effects can be separated and we obtain a system of deterministic equations with respect to Hermite-Fourier coefficients. One important property of the system is that all the statistical moments of the solution to the Burger's equation can be computed using the solution of the system only. Thus LW with WCE presents an alternative to computing moments by the Monte Carlo method (MC). It has been numerically demonstrated that LW with WCE approach is equally accurate but substantially faster than MC at least for certain classes of initial conditions.

References

A. Bensoussan and R. Temam, “Équations stochastiques du type Navier-Stokes,” Journal of Functional Analysis, vol. 13, no. 2, pp. 195–222, 1973.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. Brzeźniak, M. Capiński, and F. Flandoli, “Stochastic partial differential equations and turbulence,” Mathematical Models & Methods in Applied Sciences, vol. 1, no. 1, pp. 41–59, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. H. Cameron and W. T. Martin, “The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals,” Annals of Mathematics. Second Series, vol. 48, pp. 385–392, 1947.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Capiński and T. Zastawniak, Mathematics for Finance, Springer Undergraduate Mathematics Series, Springer, London, 2003.
View at: Zentralblatt MATH | MathSciNet
A. J. Chorin, “Hermite expansions in Monte-Carlo computation,” Journal of Computational Physics, vol. 8, no. 3, pp. 472–482, 1971.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. J. Chorin, “Gaussian fields and random flow,” Journal of Fluid Mechanics, vol. 63, no. 1, pp. 21–32, 1974.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. C. Crow and G. H. Canavan, “Relationship between a Wiener-Hermite expansion and an energy cascade,” Journal of Fluid Mechanics, vol. 41, pp. 387–403, 1970.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. De Bouard, A. Debussche, and Y. Tsutsumi, “Periodic solutions of the Korteweg-de Vries equation driven by white noise,” SIAM Journal on Mathematical Analysis, vol. 36, no. 3, pp. 815–855, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. E and Ya. G. Sinaĭ, “New results in mathematical and statistical hydrodynamics,” Russian Mathematical Surveys, vol. 55, no. 4, pp. 635–666, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer Finance, Springer, New York, 2005.
View at: MathSciNet
J. E. Gentle, Random Number Generation and Monte Carlo Methods, Statistics and Computing, Springer, New York, 2003.
View at: Zentralblatt MATH | MathSciNet
R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991.
View at: Zentralblatt MATH | MathSciNet
J. Glimm, S. Hou, H. Kim et al., “Risk management for petroleum reservoir production: a simulation-based study of prediction,” Computational Geosciences, vol. 5, no. 3, pp. 173–197, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Glimm, S. Hou, H. Kim, D. Sharp, and K. Ye, “A probability model for errors in the numerical solutions of a partial differential equation,” Computational Fluid Dynamics Journal, vol. 9, pp. 485–493, 2000.
View at: Google Scholar
J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer, New York, 2nd edition, 1987.
View at: MathSciNet
T. Y. Hou, H. Kim, B. Rozovskii, and H. Zhou, “Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics,” HERMIS, vol. 4, pp. 1–14, 2003.
View at: Google Scholar
M. Jardak, C.-H. Su, and G. E. Karniadakis, “Spectral polynomial chaos solutions of the stochastic advection equation,” Journal of Scientific Computing, vol. 17, no. 1–4, pp. 319–338, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. H. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Physics of Fluids, vol. 11, no. 5, pp. 945–953, 1968.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
E. Kreyszig, Introductory Functional Analysis with Applications, Wiley Classics Library, John Wiley & Sons, New York, 1989.
View at: Zentralblatt MATH | MathSciNet
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, vol. 6 of Course of Theoretical Physics, Pergamon Press, London, 1959.
View at: MathSciNet
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
View at: Zentralblatt MATH | MathSciNet
J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer Series in Statistics, Springer, New York, 2001.
View at: Zentralblatt MATH | MathSciNet
P.-A. Meyer, Quantum Probability for Probabilists, vol. 1538 of Lecture Notes in Mathematics, Springer, Berlin, 1993.
View at: Zentralblatt MATH | MathSciNet
R. Mikulevicius and B. L. Rozovskii, “Linear parabolic stochastic PDEs and Wiener chaos,” SIAM Journal on Mathematical Analysis, vol. 29, no. 2, pp. 452–480, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Mikulevicius and B. L. Rozovskii, “Martingale problems for stochastic PDE's,” in Stochastic Partial Differential Equations: Six Perspectives, vol. 64 of Math. Surveys Monogr., pp. 243–325, American Mathematical Society, Rhode Island, 1999.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. Mikulevicius and B. L. Rozovskii, “Stochastic Navier-Stokes equations for turbulent flows,” SIAM Journal on Mathematical Analysis, vol. 35, no. 5, pp. 1250–1310, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. N. Milstein and M. V. Tretyakov, “Evaluation of conditional Wiener integrals by numerical integration of stochastic differential equations,” Journal of Computational Physics, vol. 197, no. 1, pp. 275–298, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, Cambridge, 1994.
View at: Zentralblatt MATH | MathSciNet
G. Sansone, Orthogonal Functions, Dover, New York, 1991.
View at: MathSciNet
M. Scalerandi, A. Romano, and C. A. Condat, “Korteweg-de Vries solitons under additive stochastic perturbations,” Physical Review E, vol. 58, no. 4, pp. 4166–4173, 1998.
View at: Publisher Site | Google Scholar
R. Spigler, “Monte Carlo-type simulation for solving stochastic ordinary differential equations,” Mathematics and Computers in Simulation, vol. 29, no. 3-4, pp. 243–251, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Taylor & Francis, Washington, DC, 1997.
D. Zhang and Z. Lu, “An efficient high-order perturbation approach for flow in random porous media via Karhunun-Loeve and polynomial expansion,” Journal of Computational Physics, vol. 194, no. 2, pp. 773–794, 2004.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2006 Hongjoong Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

207

Downloads

749

Citations