Hyperbolic divergence cleaning for the MHD equations

By Andreas Dedner, Friedemann Kemm, Dietmar Kröner, Claus-Dieter Munz, T. Schnitzer, Matthias Wesenberg. 2002

1,165 citations recorded by the source corpus.

Subfields: Computational Mechanics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/158259f58bbf61b860a329633154ded154169275

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In Computational Mechanics

In Nuclear and High Energy Physics

Citations in the corpus

Beyond ideal MHD: towards a more realistic modelling of relativistic astrophysical plasmas (2008, methodology): “This approach, named hyperbolic divergence cleaning in the context of ideal MHD (Dedner et al. 2002), was proposed as a simple way of solving the Maxwell equations and enforcing the conservation of the divergence-free condition for the magnetic field.”
High- and Low-Velocity Magnetized Outflows in the Star Formation Process in a Gravitationally Collapsing Cloud (2008, positive): “We adopted the hyperbolic divergenceB cleaningmethod of Dedner et al. (2002).”
An unsplit Godunov method for ideal MHD via constrained transport (2005, methodology): “Examples of this approach include the eight-wave scheme [26] and the GLM-MHD scheme [12].”
Adaptive numerical algorithms in space weather modeling (2012, methodology): “Finally, in the hyperbolic/parabolic cleaning method of [63], a new scalar field is added to the MHD variables that couples the induction equation to the ∇ · B error:”
MHD simulations of jet acceleration from Keplerian accretion disks - The effects of disk resistivity (2007, methodology): “To control the solenoidality of the magnetic field (∇·B = 0), the eight-wave approach (Powell et al. 1999) was used, which is known for simply advecting the monopoles, while a parabolic diffusion operator (Marder 1987; see also Dedner et al. 2002) was added to the induction equation to diffuse them.”
Electrodynamics of black hole magnetospheres (2004, methodology): “There have been many attempts to find a cure for this ‘div B problem’, though no perfect solution seems to have been found so far (see the review in Dedner et al. 2002).”
Smoothed Particle Hydrodynamics in Astrophysics (2010, methodology): “To this extent, different approaches have been investigated in the literature, involving periodic field cleaning techniques (e.g. Dedner et al., 2002), formulations of the equations that ‘diffuse away’ the∇ ·B 6= 0 terms if they should arise, or the use of the Euler potentials or the vector…”
Coupling of a global MHD code and an inner magnetospheric model: Initial results (2004, methodology): “[Dedner et al., 2003], the projection scheme [Brackbill and Barnes, 1980], and a conservative form of the constrained transport scheme extended to adaptive grids”
Smoothed particle hydrodynamics and magnetohydrodynamics (2010, neutral): “[23] suggested a hyperbolic cleaning method for the MHD equations, given by dB dt 1⁄4 Bðr vÞ þ ðB rÞv rw; ð150Þ”
The Magnetic Connection between the Convection Zone and Corona in the Quiet Sun (2007, methodology): “This approach corresponds to solving the constrained MHD system of equations using an implicit parabolic correction (see Dedner et al. 2002) and acts to dissipate any local divergence errors out of the domain given an appropriate transmitting external boundary condition.”
Athena: A New Code for Astrophysical MHD (2008, background): “…a nonconservative formulation that allows propagation and damping of errors in the constraint (Powell 1994; Falle et al. 1998; Powell et al. 1999; Dedner et al. 2002), or some form of the constrained transport (CT) algorithm of Evans & Hawley (1988) to enforce the constraint (Dai & Woodward…”
RADIATION MAGNETOHYDRODYNAMIC SIMULATIONS OF PROTOSTELLAR COLLAPSE: PROTOSTELLAR CORE FORMATION (2012, methodology): “To numerically satisfy the solenoidal constraint (4), we adopt the mixed correction based on the generalized Lagrangian multiplier approach proposed by Dedner et al. (2002).”
An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics: a staggered dissipation-control differencing algorithm (2006, background): “Recently, there also have been other approaches [31, 48] to extend the 8-wave schemes that manifest the ∇ ·B source term.”
On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws (2011, methodology): “used for divergence cleaning, see [6], to satisfy the constraint ∇·~B = 0 in multiple space dimensions.”
RADIATION MAGNETOHYDRODYNAMIC SIMULATIONS OF PROTOSTELLAR COLLAPSE: NONIDEAL MAGNETOHYDRODYNAMIC EFFECTS AND EARLY FORMATION OF CIRCUMSTELLAR DISKS (2015, methodology): “In the x-direction, we adopt the fixed boundary condition, which may violate the solenoidal constraint, but the Bdiv error is cleaned by the mixed cleaning scheme (Dedner et al. 2002) and does not affect the calculations.”
A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes (2008, methodology): “In this article we use the hyperbolic version of the generalized Lagrangian multiplier (GLM) divergence cleaning approach proposed in [23].”