Abstract
An efficient algorithm is proposed and studied for computing flow ensembles of incompressible magnetohydrodynamic (MHD) flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through a clever algorithm (adapted from a breakthrough idea of Jiang and Layton [23]) that, at each time step, uses the same matrix for each of the J systems solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Elsässer variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS 1522191
Funding statement: The second author was partially supported by NSF grant DMS 1522191.
References
[1] Akbas M., Kaya S., Mohebujjaman M. and Rebholz L., Numerical analysis and testing of a fully discrete, decoupled penalty-projection algorithm for MHD in Elsässer variable, Int. J. Numer. Anal. Model. 13 (2016), no. 1, 90–113. Search in Google Scholar
[2] Arnold D. and Qin J., Quadratic velocity/linear pressure Stokes elements, Advances in Computer Methods for Partial Differential Equations VII, IMACS, New Brunswick (1992), 28–34. Search in Google Scholar
[3] Barleon L., Casal V. and Lenhart L., MHD flow in liquid-metal-cooled blankets, Fusion Engrg. Des. 14 (1991), 401–412. 10.1016/B978-0-444-88508-1.50161-4Search in Google Scholar
[4] Barrow J. D., Maartens R. and Tsagas C. G., Cosmology with inhomogeneous magnetic fields, Phys. Rep. 449 (2007), 131–171. 10.1016/j.physrep.2007.04.006Search in Google Scholar
[5] Bodenheimer P., Laughlin G. P., Rozyczka M. and Yorke H. W., Numerical Methods in Astrophysics, Ser. Astron. Astrophys., Taylor & Francis, New York, 2007. 10.1201/9781420011869Search in Google Scholar
[6] Brenner S. C. and Scott L. R., The Mathematical Theory of Finite Element Methods, Texts Appl. Math. 15, Springer, New York, 2008. 10.1007/978-0-387-75934-0Search in Google Scholar
[7] Carney M., Cunningham P., Dowling J. and Lee C., Predicting probability distributions for surf height using an ensemble of mixture density networks, Proceedings of the 22nd International Conference on Machine Learning (ICML ’05), ACM, New York (2005), 113–120. 10.1145/1102351.1102366Search in Google Scholar
[8] Davidson P. A., An Introduction to Magnetohydrodynamics, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2001. 10.1017/CBO9780511626333Search in Google Scholar
[9] Dormy E. and Soward A. M., Mathematical Aspects of Natural Dynamos. Selected by Grenoble Sciences, Fluid Mech. Astrophy. Geophys. 13, CRC Press, Boca Raton, 2007. 10.1201/9781420055269Search in Google Scholar
[10] Font J. A., Gerneral relativistic hydrodynamics and magnetohydrodynamics: Hyperbolic system in relativistic astrophysics, Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin (2008), 3–17. 10.1007/978-3-540-75712-2_1Search in Google Scholar
[11] Freitas C. J., The issue of numerical uncertainty, Appl. Math. Model. 26 (2002), 237–248. 10.1016/S0307-904X(01)00058-0Search in Google Scholar
[12] Ghanem R. and Spano P., Stochastic Finite Elements: A Spectral Approach, Dover Publications, Mineola, 2003. Search in Google Scholar
[13] Giraldo Osorio J. D. and Garcia Galiano S. G., Building hazard maps of extreme daily rainy events from PDF ensemble, via REA method, on Senegal river basin, Hydrology Earth Syst. Sci. 15 (2011), 3605–3615. 10.5194/hess-15-3605-2011Search in Google Scholar
[14] Girault V. and Raviart P.-A., Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986. 10.1007/978-3-642-61623-5Search in Google Scholar
[15] Gunzburger M., Jiang N. and Schneier M., An ensemble-proper orthogonal decomposition method for the nonstationary Navier–Stokes equations, preprint 2016, http://arxiv.org/abs/1603.04777. 10.1137/16M1056444Search in Google Scholar
[16] Hashizume H., Numerical and experimental research to solve MHD problem in liquid blanket system, MHD flow in liquid-metal-cooled blankets, Fusion Engrg. Des. 81 (2006), 1431–1438. 10.1016/j.fusengdes.2005.08.086Search in Google Scholar
[17] Hecht F., New development in freefem++, J. Numer. Math. 20 (2012), 251–266. 10.1515/jnum-2012-0013Search in Google Scholar
[18] Heister T., Mohebujjaman M. and Rebholz L., Decoupled, unconditionally stable, higher order discretizations for MHD flow simulation, J. Sci. Comput. (2016), 10.1007/s10915-016-0288-4. 10.1007/s10915-016-0288-4Search in Google Scholar
[19] Heywood J. G. and Rannacher R., Finite-element approximation of the nonstationary Navier–Stokes problem part IV: Error analysis for second-order time discretization, SIAM J.Numer. Anal. 27 (1990), 353–384. 10.1137/0727022Search in Google Scholar
[20] Hillebrandt W. and Kupka F., Interdisciplinary Aspects of Turbulence, Lecture Notes in Phys. 756, Springer, Berlin, 2009. 10.1007/978-3-540-78961-1Search in Google Scholar
[21] Jiang N., A higher order ensemble simulation algorithm for fluid flows, J. Sci. Comput. 64 (2015), 264–288. 10.1007/s10915-014-9932-zSearch in Google Scholar
[22] Jiang N., A second order ensemble method based on a blended BDF timestepping scheme for time dependent Navier–Stokes equations, Numer. Methods Partial Differential Equations (2016), 10.1002/num.22070. 10.1002/num.22070Search in Google Scholar
[23] Jiang N. and Layton W., An algorithm for fast calculation of flow ensembles, Int. J. Uncertain. Quantif. 4 (2014), 273–301. 10.1615/Int.J.UncertaintyQuantification.2014007691Search in Google Scholar
[24] Jiang N. and Layton W., Numerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion, Numer. Methods Partial Differential Equations 31 (2015), 630–651. 10.1002/num.21908Search in Google Scholar
[25] John V., Linke A., Merdon C., Neilan M. and Rebholz L., On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., to appear. 10.1137/15M1047696Search in Google Scholar
[26] Layton W., Introduction to the Numerical Analysis of Incompressible Viscous Flows, SIAM, Philadelphia, 2008. 10.1137/1.9780898718904Search in Google Scholar
[27] Le Maître O. P. and Knio O. M., Spectral Methods for Uncertainty Quantification, Springer, Dordrecht, 2010. 10.1007/978-90-481-3520-2Search in Google Scholar
[28] Lewis J. M., Roots of ensemble forecasting, Monthly Weather Rev. 133 (2005), 1865–1885. 10.1175/MWR2949.1Search in Google Scholar
[29] Martin W. J. and Xue M., Sensitivity analysis of convection of the 24 May 2002 IHOP case using very large ensembles, Monthly Weather Rev. 134 (2006), 192–207. 10.1175/MWR3061.1Search in Google Scholar
[30] Neda M., Takhirov A. and Waters J., Ensemble calculations for time relaxation fluid flow models, Numer. Methods Partial Differential Equations 32 (2016), no. 3, 757–777. Search in Google Scholar
[31] Palmer T. N. and Leutbecher M., Ensemble forecasting, J. Comput. Phys. 227 (2008), 3515–3539. 10.1016/j.jcp.2007.02.014Search in Google Scholar
[32] Punsly B., Black hole Gravitohydromagnetics, 2nd ed., Astrophys. Space Sci. Libr. 355, Springer, Berlin, 2008. 10.1007/978-3-540-76957-6Search in Google Scholar
[33] Smolentsev S., Moreau R., Buhler L. and Mistrangelo C., MHD thermofluid issues of liquid-metal blankets: Phenomena and advances, Fusion Engrg. Des. 85 (2010), 1196–1205. 10.1016/j.fusengdes.2010.02.038Search in Google Scholar
[34] Trenchea C., Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows, Appl. Math. Lett. 27 (2014), 97–100. 10.1016/j.aml.2013.06.017Search in Google Scholar
[35] Zhang S., A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp. 74 (2005), 543–554. 10.1090/S0025-5718-04-01711-9Search in Google Scholar
© 2017 by De Gruyter
