Upwind Finite-Volume Solution of Stochastic Burgers’ Equation

Affiliation(s)

Department of Engineering Mathematics & Physics, Engineering Faculty, Cairo University, Giza, Egypt.

Department of Engineering Mathematics & Physics, Faculty of Engineering, Fayoum University, Fayoum, Egypt.

Department of Engineering Mathematics & Physics, Engineering Faculty, Cairo University, Giza, Egypt.

Department of Engineering Mathematics & Physics, Faculty of Engineering, Fayoum University, Fayoum, Egypt.

ABSTRACT

In this paper, a stochastic finite-volume solver based on polynomial chaos expansion is developed. The upwind scheme is used to avoid the numerical instabilities. The Burgers’ equation subjected to deterministic boundary conditions and random viscosity is solved. The solution uncertainty is quantified for different values of viscosity. Monte-Carlo simulations are used to validate and compare the developed solver. The mean, standard deviation and the probability distribution function (p.d.f) of the stochastic Burgers’ solution is quantified and the effect of some parameters is investigated. The large sparse linear system resulting from the stochastic solver is solved in parallel to enhance the performance. Also, Monte-Carlo simulations are done in parallel and the execution times are compared in both cases.

In this paper, a stochastic finite-volume solver based on polynomial chaos expansion is developed. The upwind scheme is used to avoid the numerical instabilities. The Burgers’ equation subjected to deterministic boundary conditions and random viscosity is solved. The solution uncertainty is quantified for different values of viscosity. Monte-Carlo simulations are used to validate and compare the developed solver. The mean, standard deviation and the probability distribution function (p.d.f) of the stochastic Burgers’ solution is quantified and the effect of some parameters is investigated. The large sparse linear system resulting from the stochastic solver is solved in parallel to enhance the performance. Also, Monte-Carlo simulations are done in parallel and the execution times are compared in both cases.

Cite this paper

M. El-Beltagy, M. Wafa and O. Galal, "Upwind Finite-Volume Solution of Stochastic Burgers’ Equation,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1818-1825. doi: 10.4236/am.2012.331247.

M. El-Beltagy, M. Wafa and O. Galal, "Upwind Finite-Volume Solution of Stochastic Burgers’ Equation,"

References

[1] O. H. Galal, W. El-Tahan, M. A. El-Tawil and A. A. Mahmoud, “Spectral SFEM Analysis of Structures with Stochastic Parameters under Stochastic Excitation,” Structural Engineering and Mechanics, Vol. 28 No. 3, 2008, pp. 281-294.

[2] O. H. Galal, “The Solution of Stochastic Linear Partial Differential Equation Using SFEM through Neumann and Homogeneous Chaos Expansions,” Ph.D. Thesis, Cairo University, Cairo, 2005.

[3] S. Rahman and H. Xu, “A Meshless Method for Computational Structure Mechanics,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 6, No. 1, 2005, pp. 41-58. doi:10.1080/15502280590888649

[4] M. Kaminski, “Stochastic Perturbation Approach to Engineering Structure Variability by the Finite Difference Method,” Journal of Sound and Vibration, Vol. 251 No. 4, 2002, pp. 651-670. doi:10.1006/jsvi.2001.3850

[5] G. Stefanou, “The Stochastic Finite Element Methods: Past, Present and Future,” Computer Methods in Applied Mechanics and Engineering, Vol. 198, No. 9-12, 2009, pp. 1031-1051. doi:10.1016/j.cma.2008.11.007

[6] M. Shinozuka and T. Nomoto, “Response Variability Due to Spatial Randomness of Material Properties,” Technical Report, Columbia University, New York, 1980.

[7] A. Henriques, J. Veiga, J. Matos and J. Delgado, “Uncertainty Analysis of Structural Systems by Perturbation Techniques,” Journal of Structural and Multidisciplinary Optimization, Vol. 35 No. 3, 2008, pp. 201-212. doi:10.1007/s00158-007-0218-z

[8] R. Ghanem and P. D. Spanos, “Stochastic Finite Elements: A Spectral Approach,” 2nd Edition, Dover, New-York, 2002.

[9] H. Panayirci, “Computational Strategies for Efficient Stochastic Finite Element Analysis of Engineering Structures,” Ph.D. Thesis, University of Innsbruck, Innsbruck, 2010.

[10] J. Hurtado, “Analysis of One Dimensional Stochastic Finite Element Using Neural Network,” Probabilistic Engineering Mechanics, Vol.17, No. 1, 2001, pp. 35-44.

[11] M. A. El-Beltagy, O. H. Galal and M. I. Wafa, “Uncertainty Quantification of a 1-D Beam Deflection Due to Stochastic Parameters,” International Conference on Numerical Analysis and Applied Mathematics, Halkidiki, 19-25 September 2011, pp. 2000-2003. doi:10.1063/1.3637007.

[12] J. D. Cole, “On a Quasilinear Parabolic Equations Occurring in Aerodynamics,” Quarterly of Applied Mathematics, Vol. 9, 1951, pp. 225-236.

[13] J. D. Logan, “An Introduction to Nonlinear Partial Differential Equations,” Wily-Interscience, New York, 1994.

[14] L. Debtnath, “Nonlinear Partial Differential Equations for Scientist and Engineers,” Birkhauser, Boston, 1997.

[15] G. Adomian, “The Diffusion-Brusselator Equation,” Computers & Mathematics with Applications, Vol. 29, No. 5, 1995, pp. 1-3. doi:10.1016/0898-1221(94)00244-F

[16] C. Fletcher, “Burgers’ Equation: A Model for All Reasons,” Numerical Solutions of Partial Differential Equations, North-Holland Pub. Co., Holland, 1982.

[17] A. B. Stephens, R. B. Kellogg and G. R. Shubin, “Uniqueness and the Cell Reynolds Number,” SIAM Journal on Numerical Analysis, Vol. 17, No. 6, 1980.

[18] O. Schenk and K. G?rtner, “Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO,” Journal of Future Generation Computer Systems, Vol. 20, No. 3, 2004, pp. 475-487. doi:10.1016/j.future.2003.07.011

[19] O. Schenk and K. G?rtner, “On Fast Factorization Pivoting Methods for Symmetric Indefinite Systems,” Electronic Transactions on Numerical Analysis, Vol. 23, 2006, pp. 158-179.

[1] O. H. Galal, W. El-Tahan, M. A. El-Tawil and A. A. Mahmoud, “Spectral SFEM Analysis of Structures with Stochastic Parameters under Stochastic Excitation,” Structural Engineering and Mechanics, Vol. 28 No. 3, 2008, pp. 281-294.

[2] O. H. Galal, “The Solution of Stochastic Linear Partial Differential Equation Using SFEM through Neumann and Homogeneous Chaos Expansions,” Ph.D. Thesis, Cairo University, Cairo, 2005.

[3] S. Rahman and H. Xu, “A Meshless Method for Computational Structure Mechanics,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 6, No. 1, 2005, pp. 41-58. doi:10.1080/15502280590888649

[4] M. Kaminski, “Stochastic Perturbation Approach to Engineering Structure Variability by the Finite Difference Method,” Journal of Sound and Vibration, Vol. 251 No. 4, 2002, pp. 651-670. doi:10.1006/jsvi.2001.3850

[5] G. Stefanou, “The Stochastic Finite Element Methods: Past, Present and Future,” Computer Methods in Applied Mechanics and Engineering, Vol. 198, No. 9-12, 2009, pp. 1031-1051. doi:10.1016/j.cma.2008.11.007

[6] M. Shinozuka and T. Nomoto, “Response Variability Due to Spatial Randomness of Material Properties,” Technical Report, Columbia University, New York, 1980.

[7] A. Henriques, J. Veiga, J. Matos and J. Delgado, “Uncertainty Analysis of Structural Systems by Perturbation Techniques,” Journal of Structural and Multidisciplinary Optimization, Vol. 35 No. 3, 2008, pp. 201-212. doi:10.1007/s00158-007-0218-z

[8] R. Ghanem and P. D. Spanos, “Stochastic Finite Elements: A Spectral Approach,” 2nd Edition, Dover, New-York, 2002.

[9] H. Panayirci, “Computational Strategies for Efficient Stochastic Finite Element Analysis of Engineering Structures,” Ph.D. Thesis, University of Innsbruck, Innsbruck, 2010.

[10] J. Hurtado, “Analysis of One Dimensional Stochastic Finite Element Using Neural Network,” Probabilistic Engineering Mechanics, Vol.17, No. 1, 2001, pp. 35-44.

[11] M. A. El-Beltagy, O. H. Galal and M. I. Wafa, “Uncertainty Quantification of a 1-D Beam Deflection Due to Stochastic Parameters,” International Conference on Numerical Analysis and Applied Mathematics, Halkidiki, 19-25 September 2011, pp. 2000-2003. doi:10.1063/1.3637007.

[12] J. D. Cole, “On a Quasilinear Parabolic Equations Occurring in Aerodynamics,” Quarterly of Applied Mathematics, Vol. 9, 1951, pp. 225-236.

[13] J. D. Logan, “An Introduction to Nonlinear Partial Differential Equations,” Wily-Interscience, New York, 1994.

[14] L. Debtnath, “Nonlinear Partial Differential Equations for Scientist and Engineers,” Birkhauser, Boston, 1997.

[15] G. Adomian, “The Diffusion-Brusselator Equation,” Computers & Mathematics with Applications, Vol. 29, No. 5, 1995, pp. 1-3. doi:10.1016/0898-1221(94)00244-F

[16] C. Fletcher, “Burgers’ Equation: A Model for All Reasons,” Numerical Solutions of Partial Differential Equations, North-Holland Pub. Co., Holland, 1982.

[17] A. B. Stephens, R. B. Kellogg and G. R. Shubin, “Uniqueness and the Cell Reynolds Number,” SIAM Journal on Numerical Analysis, Vol. 17, No. 6, 1980.

[18] O. Schenk and K. G?rtner, “Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO,” Journal of Future Generation Computer Systems, Vol. 20, No. 3, 2004, pp. 475-487. doi:10.1016/j.future.2003.07.011

[19] O. Schenk and K. G?rtner, “On Fast Factorization Pivoting Methods for Symmetric Indefinite Systems,” Electronic Transactions on Numerical Analysis, Vol. 23, 2006, pp. 158-179.