Numerical experiments areused to illustrate the theoretical results.
📌 3 Key Takeaways
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Objective & Hypothesis:Thermally coupled magneto-hydrodynamics (MHD) studies the dynamics ofelectro-magnetically and thermally driven flows, involving MHD equations cou-pled with heat equation.
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Key Findings & Data:We introduce a partitioned method that allows oneto decouple the MHD equations from the heat equation at each time step andsolve them separately.
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Conclusion & Impact:Numerical experiments areused to illustrate the theoretical results.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
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.
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Cite This Article
RAVİNDRAN, S. (2022). A decoupled Crank-Nicolson time-stepping scheme for thermally coupled magneto-hydrodynamic system. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), Cilt: 8 - Sayı: 1, 43–62. https://ajindex.com/articles/440643
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.