On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method

By Pasquale Londrillo, L. Del Zanna. 2003

246 citations recorded by the source corpus.

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 [Δ ċ B]num = 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 Roetype positive: scheme by Lin and Lax [J. Comput. Fluid Dyn. 5 (1996) 133], and both the second- and third-order versions of a central-type MHD scheme presented by Londrillo and Del Zanna [Astrophys. J. 530 (2000) 508], where the basic UCT strategies have been first outlined.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/b6a0ca64efcfaa9a634f062b33de671db7d1471d

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

An unsplit Godunov method for ideal MHD via constrained transport (2005, methodology): “In recent years a variety of numerical algorithms for multidimensional magne-tohydrodynamics (MHD) based on Godunov’s method have been developed [3,9,10,16,23,25,28].”
The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project (2019, background): “ECHO The origin of the Eulerian Conservative High - Order (ECHO) code dates back to the year 2000 (Londrillo & Del Zanna 2000, 2004), when it was first proposed as a shockcapturing scheme for classical MHD based on high-order, finite-difference reconstruction routines, one-wave or two-wave Riemann…”
An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics: a staggered dissipation-control differencing algorithm (2006, unknown): “The next test to consider is the rotor problem which was originally studied by Balsara & Spicer [9], and also by many others [11, 55, 56, 60, 88].”