Abstract
In this paper, we construct a second order algorithm based on the spectral deferred correction method for the time-dependent magnetohydrodynamics flows at a low magnetic Reynolds number. We present a complete theoretical analysis to prove that this algorithm is unconditionally stable, consistent and second order accuracy. Finally, two numerical examples are given to illustrate the convergence and effectiveness of our algorithm.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alfvén, H.: Existence of electromagnetic-hydrodynamic waves. Nature 150, 405–406 (1942)
Barleon, L., Casal, V., Lenhart, L.: MHD Flow in liquid-metal-cooled blankets. Fusion Eng. Des. 14, 401–412 (1991)
Davidson, P. A.: Magnetohydrodynamics in material processing. Annu. Rev. Fluid Mech. 31, 273–300 (1999)
Lin, T. F., Gilbert, J. B., Kossowsky, R.: Sea-water magnetohydrodynamic propulsion for next-generation undersea vehicles. Pennsylvania State University State College Applied Research Laboratory (1990)
Gerbeau, J. F., Bris, C. L., Lelièvre, T.: Mathmatical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press (2006)
Adams, R. A.: Sobolev Spaces. Academic Press (2003)
Gunzburger, M. D., Meir, A. J., Peterson, J.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible Magnetohydrodynamics. Math. Comput. 56(194), 523–563 (1991)
Yuksel, G., Ingram, R.: Numerical analysis of a finite element Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers. Int. J. Numer. Anal. Model. 10(1), 74–98 (2013)
Meir, A. J., Schmidt, P. G.: Analysis and numerical approximation of stationary MHD flow problem with nonideal boundary. SIAM J. Numer. Anal. 36(4), 1304–1332 (1999)
Meir, A. J., Schmidt, P. G.: Variational methods for stationary MHD flow under natural interface conditions. Nonlinear Anal. 26(4), 659–689 (1996)
Meir, A. J.: The equations of stationary, incompressible magnetohydynamics with mixed boundary conditions. Comput. Math. Appl. 25(12), 13–29 (1993)
Layton, W. J., Meir, A. J., Schmidt, P. G.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198–210 (1997)
Schmidt, P. G.: A Galerkin method for time-dependent MHD flow with nonideal boundaries. Commun. Appl. Anal. 3(3), 383–398 (1999)
Layton, W. J., Tran, H., Trenchea, C.: Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows. Numer. Meth. Part D.E. 30(4), 1083–1102 (2014)
Yuksel, G., Isik, O. R.: Numerical analysis of Backward-Euler discretization for simplified magnetohydynamic flows. Applied Mathematical Modelling. 39, 1889–1898 (2015)
Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT. 40(2), 241–266 (2000)
Minion, M. L.: Semi-implicit projection methods for incompressible flow based on spectral deferred corrections. Appl. Numer. Math. 48(3-4), 369–387 (2004)
Minion, M. L.: Semi-Implicit Projection methods for ordinary differential equations. Comm. Math. Sci. 1(3), 471–500 (2003)
Wilson, N., Labovsky, A., Trenchea, C.: High accuracy method for magnetohydynamic system in Elsässer variables. Computational Methods in Applied Mathematics 15(1), 97–110 (2014)
He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA Journal of Numerical Analysis. dru015 (2014)
Roberts, P. H.: An Introduction to Magnetohydynamics. Elsevier, USA (1967)
Hecht, F., Pironneau, O.: Freefem++. Webpage: http://www.freefem.org
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Silas Alben
Supported by NSFC (Grant No. 11171269 and 11571274).
Rights and permissions
About this article
Cite this article
Rong, Y., Hou, Y. & Zhang, Y. Numerical analysis of a second order algorithm for simplified magnetohydrodynamic flows. Adv Comput Math 43, 823–848 (2017). https://doi.org/10.1007/s10444-016-9508-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-016-9508-6