The solution of hyperbolic equations in magnetohydrodynamics
It is shown that the solution of the hyperbolic MHD evolution equations for large but finite values of the CFL number S can conveniently be treated as the solution of a sequence of perturbed elliptic equilibrium problems, the perturbing term ρ Du/Dt vanishing as S → ∞. The implicit method developed by Hain and used in the 1DMHD Hain-Roberts code is re-examined from this point of view for a Lagrangian difference scheme. Oscillations of the Lagrangian mesh with period xxx 2Δt can be excited by non-linear 1 coupling terms but need not affect the solution provided that due care is taken especially in the control of At by monitoring the rate of change of the physical variables. Straightforward use of the implicit Crank-Nicholson scheme with θ = ½ leads to mesh oscillations which are undamped and could build up to very large velocity fluctuations δu ,due to energy equipartition, and it is therefore recommended that damping should be introduced by choosing θ > ½. Details are presented for 1D geometry but it is believed that the method could also be used for studying the 2D and 3D evolution of non-linear MHD instabilities.
KeywordsInertial Term Implicit Method Energy Equipartition Lagrangian Mesh Adiabatic Oscillation
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