A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system
A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system
Thermally coupled magneto-hydrodynamics (MHD) studies the dynamics ofelectro-magnetically and thermally driven flows, involving MHD equations cou-pled with heat equation. We introduce a partitioned method that allows oneto decouple the MHD equations from the heat equation at each time step andsolve them separately. The extrapolated Crank-Nicolson time-stepping schemeis used for time discretization while mixed finite element method is used forspatial discretization.We derive optimal order error estimates in suitable normswithout assuming any stability condition or restrictions on the time step size.We prove the unconditional stability of the scheme. Numerical experiments areused to illustrate the theoretical results.
___
- Barleon, L., Casal, V., and Lenhart, L. (1991). MHD
flow in liquid-metal-cooled blankets. Fusion Engineer-
ing and Design, 14(3-4), 401-412.
- Davidson, P.A. (1999). Magneto-hydrodynamics in
material processing. Annual Review of Fluid Mechan-
ics, 31 , 273-300.
- DiBenedetto, E. (1993). Degenerate Parabolic Equa-
tions, Springer-Verlag, New York.
- El-Kaddah, N., Patel, A.D. and Natarajan, T.T.
(1995). The electromagnetic filtration of molten alu-
minum using an induced-current separator. Journal of
the Minerals, Metals and Materials Society, 47, 46-49.
- Ervin, V.J. and Heuer, N. (2004). Approximation
of time-dependent, viscoelastic fluid flow: Crank-
Nicolson, finite element approximation. Numer. Meth-
ods Partial Differential Equations, 20(2), 248-283.
- Girault, V. and Raviart, P.A. (1986). Finite Element
Method for Navier-Stokes Equations, Springer, Berlin.
- Gerbeau, J.F. (2000). A stabilized finite element
methods for incompressible methods magnetohydro-
dynamic equations. Numer. Math., 87, 83-111.
- Guermond, J. L. and Minev, P.D. (2003). Mixed finite
element approximation of an MHD problem involv-
ing conducting and insulating regions: The 3D case.
Numer. Methods Partial Differential Equations, 19(6),
709-731.
- Gunzburger, M.D., Meir, A.J. and Peterson, J. (1991).
On the existence, uniqueness, and finite element ap-
proximation of solutions of the equations of station-
ary, incompressible magnetohydrodynamics. Math.
Comp., 56(194), 523-563.
- Gunzburger, M.D. and Peterson, J. (1983). On con-
forming finite element methods for the inhomogeneous
stationary Navier-Stokes equations. Numer. Math.,
42, 173-194.
- Hashizume, H. (2006). Numerical and experimental
research to solve MHD problem in liquid-blanket sys-
tem. Fusion Engineering and Design, 81(8), 1431-
1438.
- Heywood, J.G. and Rannacher, R. (1990). Finite el-
ement approximation of the non-stationary Navier-
Stokes problem, Part IV: Error for second order time
discretization. SIAM Journal of Numerical Analysis,
27(2), 353-384.
- Hughes, W. F. and Young, F.J. (1966). The electro-
magnetodynamicos of fluids. Wiley, New York,
- Ingram, R. (2013). A new linearly extrapolated Crank-
Nicolson time-stepping scheme for the Navier-Stokes
equations. Mathematics of Computation, 82(284),
- Julien, K., Knobloch, E. and Tobias, S. (1999).
Strongly nonlinear magnetoconvection in three dimen-
sions. Physica D, 128, 105-129.
- Kherief, K. (1984). Quelques proprietes des equations
de la magnetohydrodynamique stationnaires et devo-
lution. Th‘ese, Universite de Paris VII.
- Layton, W., Tran, H. and Trenchea, C. (2013). Stabil-
ity of partitioned methods for magnetohydrodynamics
flows at small magnetic Reynolds number. Contemp.
Math., 586, 231-238.
- Lin, T.F., Gilbert, J.B. and Roy, G.D. (1991). Analy-
ses of magnetohydrodynamic propulsion with seawa-
ter for underwater vehicles. Journal of Propulsion and
Power, 7(6), 1081-1083.
- Lifschitz, A.E. (1989). Magnetohydrodynamics and
Spectral Theory. Kluwer, Dordrecht.
- Meir, A.J. and Schmidt, P.G. (1996). Variational
methods for stationary MHD flow under natural inter-
face conditions. Nonlinear Analysis, 26(4), 659-689.
- Meir, A.J. and Schmidt, P.G. (1999). Analysis and nu-
merical approximation of a stationaryMHD flow prob-
lem with nonideal boundary. SIAM J. Numer. Anal.,
36(4), 1304-1332.
- Monk, P. (2003). Finite Element Methods for
Maxwell’s Equations. Oxford University Press.
- Prohl, A. (2008). Convergent finite element discretiza-
tions of the nonstationary incompressible magneto-
hydrodynamics system. M2AN Math. Model. Numer.
Anal., 42, 1065-1087.
- Rappaz, J. and Touzani, R. (1991). On a two-
dimensional magnetohydrodynamic problem. I) mod-
elling and analysis. Modelisation mathematique et
Analyse numerique, 26(2), 347-364.
- Ravindran, S.S. (2015). An extrapolated second or-
der backward difference time-stepping scheme for
the magnetohydrodynamics system. Numerical Func-
tional Analysis and Optimization, submitted.
- Schmidt, P.G. (1999). A Galerkin method for time-
dependent MHD flow with nonideal boundaries. Com-
mun. Appl. Anal., 3(3), 383-398.
- Schonbek, M.E., Schonbek, T.P. and Soli, E. (1996).
Large-time behaviour of solutions to the magnetohy-
drodynamics equations. Math. Ann., 304, 717-756.
- Scott, L.R. and Zhang, S. (1990). Finite element inter-
polation of nonsmooth functions satisfying boundary
conditions. Math. Comp., 54, 483-493.
- Sermange, M. and Temam,R. (1983). Some math-
ematical questions related to the MHD equations.
Comm. Pure Appl. Math., 36, 635-664.
- Shercliff, J.A. (1965). A textbook of magnetohydrody-
namics. Pergamon Press, Oxford.
- Smolentsev, S., Moreau, R., Buhler, L. and Mis-
trangelo, C. (2010). MHD thermofluid issues of liq-
uid metal blankets: Phenomena and advances. Fusion
Engineering and Design, 85(7-9), 1196-1205.
- Spitzer, K.H., Dubke, M. and Schwertfeger, K. (1986).
Rotational electromagnetic stirring in continuous cast-
ing of round strands. Metallurgical Transactions B, 17,
119-131.
- Tone, F. (2009). On the long-time H2-stability of the
implicit Euler scheme for the 2D magnetohydrody-
namics equations. J. Sci. Comput., 38, 331-348.
- Utech, H.P. and Flemings, M.C. (1967). Thermal con-
vection in metal-crystal growth [in tin and aluminium-
copper alloys] : Effect of a magnetic field, Proceedings
of Internat. Conf. on Crystal Growth, Boston, June
1966, J. Phys. Chem. Solids, (Suppl. 1), 651-658.
- WalkerJ, S. (1980). Large interaction parameter mag-
netohydrodynamics and applications in fusion reactor
technology. Fluid Mechanics in Energy Conversion (J.
Buckmaster, ed.), SIAM, Philadelphia.