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On the Divergence-Free Condition in Godunov-Type Schemes for Ideal Magnetohydrodynamics: the Upwind Constrained Transport Method

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arxiv astro-ph/0310183 v1 pith:WOXAXZBH submitted 2003-10-07 astro-ph

On the Divergence-Free Condition in Godunov-Type Schemes for Ideal Magnetohydrodynamics: the Upwind Constrained Transport Method

classification astro-ph
keywords constrainedfieldgeneralschemestransportupwinddesigndivergence-free
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We present a general framework to design Godunov-type schemes for multidimensional ideal magnetohydrodynamic (MHD) systems, having the divergence-free relation and the related properties of the magnetic field B as built-in conditions. Our approach mostly relies on the 'Constrained Transport' (CT) discretization technique for the magnetic field components, originally developed for the linear induction equation, which assures div(B)=0 and its preservation in time to within machine accuracy in a finite-volume setting. We show that the CT formalism, when fully exploited, can be used as a general guideline to design the reconstruction procedures of the B vector field, to adapt standard upwind procedures for the momentum and energy equations, avoiding the onset of numerical monopoles of O(1) size, and to formulate approximate Riemann solvers for the induction equation. This general framework will be named here 'Upwind Constrained Transport' (UCT). To demonstrate the versatility of our method, we apply it to a variety of schemes, which are finally validated numerically and compared: a novel implementation for the MHD case of the second order Roe-type positive scheme by Liu and Lax (J. Comp. Fluid Dynam. 5, 133, 1996), and both the second and third order versions of a central-type MHD scheme presented by Londrillo and Del Zanna (Astrophys. J. 530, 508, 2000), where the basic UCT strategies have been first outlined.

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