Second order unconditionally convergent and energy stable linearized scheme for MHD equations
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Second order unconditionally convergent and energy stable linearized scheme for MHD equations Guo-Dong Zhang1 · Jinjin Yang2 · Chunjia Bi1
Received: 31 October 2016 / Accepted: 13 July 2017 © Springer Science+Business Media, LLC 2017
Abstract In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L2 and H 1 fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/AdamsBashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.
Communicated by: Long Chen Guodong Zhang is supported by National Science Foundation of China (11601468); Chunjia Bi is supported by National Science Foundation of China (11571297) and Shandong Province Natural Science Foundation (ZR2014AM003). Guo-Dong Zhang
[email protected] Jinjin Yang [email protected] Chunjia Bi [email protected] 1
School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China
2
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
G.-D. Zhang et al.
Keywords MHD equations · Second order time discrete scheme · Unconditional convergence · Unconditional energy stability · Error estimates · Finite element method · High physical parameters Mathematics Subject Classification (2010) 65M60 · 65M12
1 Introduction We study the following incompressible d(d = 2, 3) dimensional MHD equations under the influence of the body force and applied currents: ⎧ ut − νu + (u · ∇)u + ∇p + sB × curlB = f , ⎪ ⎪ ⎨ divu = 0, (1.1) B ⎪ t + ηcurlcurlB − curl(u × B) = g, ⎪ ⎩ divB = 0, for (x, t) ∈ × (0, T ) with ⊂ Rd (d = 2, 3) and a fixed T ∈ (0, ∞). Here, is a connected, bounded domain which is either convex or has a C 1,1 boundary, u denotes the velocity field, p the pressure, B the magnetic field, f the known body force, g the known applied current with divg = 0. The physical parameters ν −1 = Re (fluid Reynolds number), η−1 = Rm (magnetic Reynolds number), the coupling coefficient s are given by Re =
UL B2 , Rm = μm σ U L, s = , μf ρμm U 2
where U is the characteristic velocity, L the characteristic length, μf the kinematic viscosity, μm the magnetic permeability, σ the electric conductivity, B the characteristic magnetic field, ρ the fluid density. The system is considered in conjunction with the following initial boundary conditions u|∂T = 0,
B · n|∂T = 0,
n × curlB|∂
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