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Journal of Scientific Computing

, Volume 79, Issue 2, pp 1271–1293 | Cite as

Ensemble Time-Stepping Algorithm for the Convection-Diffusion Equation with Random Diffusivity

  • Ning Li
  • Joseph Fiordilino
  • Xinlong FengEmail author
Article
  • 75 Downloads

Abstract

In this paper, we develop two ensemble time-stepping algorithms to solve the convection-diffusion equation with random diffusion coefficients, forcing terms and initial conditions based on the pseudo-spectral stochastic collocation method. The key step of the pseudo-spectral stochastic collocation method is to solve a number of deterministic problems derived from the original stochastic convection-diffusion equation. In general, a common way to solve the set of deterministic problems is by using the backward differentiation formula, which requires us to store the coefficient matrix and right-hand-side vector multiple times, and solve them one by one. However, the proposed algorithm only need to solve a single linear system with one shared coefficient matrix and multiple right-hand-side vectors, reducing both storage required and computational cost of the solution process. The stability and error analysis of the first- and second-order ensemble time-stepping algorithms are provided. Several numerical experiments are presented to confirm the theoretical analyses and verify the feasibility and effectiveness of the proposed method.

Keywords

Ensemble time-stepping algorithm Convection-diffusion equation Random diffusivity Stochastic collocation method Backward differentiation formula 

Mathematics Subject Classification

60H15 35R60 65M12 65M60 

Notes

Acknowledgements

The authors thank Professor William Layton for his helpful comments and suggestions.

Funding:

Ning Li is supported by China Scholarship Council Grant 201707010004, Joseph Fiordilino is supported by the DoD SMART Scholarship and partially supported by NSF Grants CBET 1609120 and DMS 1522267, Xinlong Feng is supported by the NSF of China (No. 11671345, No. 11362021).

References

  1. 1.
    Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Xiu, D.: Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2, 293–309 (2007)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Xiu, D.: Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys. 5, 242–272 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Xiu, D.: Stochastic collocation methods: a survey. In: Ghanem, R., Higdon, D., Owhadi, H. (eds.) Handbook of Uncertainty Quantification. Springer, Cham (2017)Google Scholar
  5. 5.
    Babuska, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45, 1005–1034 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52, 317–355 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhang, G., Gunzburger, M.: Error analysis of a stochastic collocation method for parabolic partial differential equations with random input data. SIAM J. Numer. Anal. 50, 1922–1940 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang, Q., Li, Z., Zhang, Z.: A sparse grid stochastic collocation method for elliptic interface problems with random input. J. Sci. Comput. 67, 262–280 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Tang, T., Zhou, T.: Recent developments in high order numerical methods for uncertainty quantification. Sci. Sin. Math. 45, 891–928 (2015)Google Scholar
  12. 12.
    Tang, T., Zhou, T.: On discrete least-squares projection in unbounded domain with random evaluations and its application to parametric uncertainty quantification. SIAM J. Sci. Comput. 36, A2272–A2295 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jakeman, J.D., Narayan, A., Zhou, T.: A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions. SIAM J. Sci. Comput. 39, A1114–A1144 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Narayan, A., Zhou, T.: Stochastic collocation on unstructured multivariate meshes. Commun. Comput. Phys. 18, 1–36 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guo, L., Narayan, A., Zhou, T., Chen, Y.: Stochastic collocation methods via \(\ell _1\) minimization using randomized quadratures. SIAM J. Sci. Comput. 39, A333–A359 (2017)CrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, L., Zheng, B., Lin, G., Voulgarakis, N.: A two-level stochastic collocation method for semilinear elliptic equations with random coefficients. J. Comput. Appl. Math. 315, 195–207 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fishman, G.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  18. 18.
    Jiang, N., Layton, W.: An algorithm for fast calculation of flow ensembles. Int. J. Uncertain. Quantif. 4, 273–301 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jiang, N., Kaya, S., Layton, W.: Analysis of model variance for ensemble based turbulence modeling. Comput. Meth. Appl. Math. 15, 173–188 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Jiang, N.: A higher order ensemble simulation algorithm for fluid flows. J. Sci. Comput. 64, 264–288 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gunzburger, M., Jiang, N., Wang, Z.: An efficient algorithm for simulating ensembles of parameterized flow problems, IMA J. Numer. Anal. (2018).  https://doi.org/10.1093/imanum/dry029
  22. 22.
    Gunzburger, M., Jiang, N., Wang, Z.: A second-order time-stepping scheme for simulating ensembles of parameterized flow problems. Comput. Methods Appl. Math. (2018).  https://doi.org/10.1515/cmam-2017-0051
  23. 23.
    Fiordilino, J.A.: A second order ensemble timestepping algorithm for natural convection. SIAM J. Numer. Anal. 56, 816–837 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fiordilino, J.A., Khankan, S.: Ensemble timestepping algorithms for natural convection. Int. J. Numer. Anal. Model. 15, 524–551 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mohebujjaman, M., Rebholz, L.: An efficient algorithm for computation of MHD flow ensembles. Comput. Methods Appl. Math. 17, 121–137 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Fiordilino, J.A.: Ensemble timestepping algorithms for the heat equation with uncertain conductivity. Numer. Meth. Partial. Differ. Eqs. 34, 1901–1916 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Luo, Y., Wang, Z.: An ensemble algorithm for numerical solutions to deterministic and random parabolic PDEs. SIAM J. Numer. Anal. 56, 859–876 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Phoon, K.K., Huang, S.P., Quek, S.T.: Simulation of second-order processes using Karhunen-Loève expansion. Comput. Struct. 80, 1049–1060 (2002)CrossRefGoogle Scholar
  29. 29.
    Zhang, D., Lu, Z.: An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Loève and polynomial expansions. J. Comput. Phys. 194, 773–794 (2004)CrossRefzbMATHGoogle Scholar
  30. 30.
    Øksendal, B.: Stochastic Differential Equations: an Introduction with Applications. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  31. 31.
    Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, New York (2007)Google Scholar
  32. 32.
    Hecht, F.: New development in freefem++. J. Numer. Math. 20, 251–265 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Information and Management of ScienceHenan Agricultural UniversityZhengzhouPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA
  3. 3.College of Mathematics and Systems ScienceXinjiang UniversityÜrümqiPeople’s Republic of China

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