MHD-kinetic hybrid code based on structure-preserving finite elements with particles-in-cell

Holderied, Florian; Possanner, Stefan; Wang, X.

doi:10.1016/j.jcp.2021.110143

Cited by 20 publications

(21 citation statements)

References 47 publications

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“…Regarding the spatial discretization, we adopt a finite‐volume/finite‐difference scheme on staggered grids that can be considered as an extention of the previous work 39 to an implicit treatment of the magnetic forces, but it can be regarded also as a first finite‐difference actuation of the theory of finite element exterior calculus (FEEC) in the discretization of the MHD equations 52,53 . In this work the choice of staggered grids is motivated mainly by two argument: the preservation of the divergence‐free condition of the magnetic field, or the velocity field in the incompressible limit; the preservation of the duality relation between the discrete differential operators.…”

Section: Semi‐implicit Discretization On Staggered Gridsmentioning

confidence: 99%

“…Regarding the spatial discretization, we adopt a finite-volume/finite-difference scheme on staggered grids that can be considered as an extention of the previous work 39 to an implicit treatment of the magnetic forces, but it can be regarded also as a first finite-difference actuation of the theory of finite element exterior calculus (FEEC) in the discretization of the MHD equations. 52,53 In this work the choice of staggered grids is motivated mainly by two argument: the preservation of the divergence-free condition of the magnetic field, or the velocity field in the incompressible limit; the preservation of the duality relation between the discrete differential operators. Dating back to Yee, 54 the use of staggered grids is known to be particularly indicated for the electric and magnetic fields, and then also for the MHD equations, Indeed, it has been shown that an electric field E defined on a mesh that is dual with respect to the mesh of the magnetic field B is the natural way of preserving the divergence-free condition up to machine error.…”

Section: Semi-implicit Discretization On Staggered Gridsmentioning

confidence: 99%

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A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

Fambri

2021

Numerical Methods in Fluids

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In this work we introduce a novel semi‐implicit structure‐preserving finite‐volume/finite‐difference scheme for the viscous and resistive equations of magneto‐hydrodynamics (VRMHD) based on an appropriate 3‐split of the governing PDE system, which is decomposed into a first convective subsystem, a second subsystem involving the coupling of the velocity field with the magnetic field, and a third subsystem involving the pressure‐velocity coupling. The nonlinear convective terms are discretized explicitly, while the remaining two subsystems accounting for the Alfvén waves and the magneto‐acoustic waves are treated implicitly. Thanks to this, the final algorithm is at least formally constrained only by a mild CFL stability condition depending on the velocity field of the pure hydrodynamic convection. To preserve the divergence‐free constraint of the magnetic field exactly at the discrete level, a proper set of overlapping dual (staggered) meshes is employed. The resulting linear algebraic systems are shown to be symmetric and therefore can be solved by means of an efficient standard matrix‐free conjugate gradient algorithm. The final scheme can be regarded as a novel shock‐capturing, conservative, and structure preserving semi‐implicit scheme for VRMHD. Several numerical tests are presented to show the main features of our novel divergence‐free semi‐implicit FV/FD solver: linear‐stability in the sense of Lyapunov is verified at a prescribed constant equilibrium solution; a second‐order of convergence is obtained for a smooth time‐dependent solution; shock‐capturing capabilities are proven against a standard set of stringent MHD shock‐problems; accuracy and robustness are verified against a non‐trivial set of two‐ and three‐dimensional MHD problems.

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Section: Semi‐implicit Discretization On Staggered Gridsmentioning

confidence: 99%

Section: Semi-implicit Discretization On Staggered Gridsmentioning

confidence: 99%

A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

Fambri

2021

Numerical Methods in Fluids

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“…Structure-preserving particle-in-cell (PIC) algorithms preserve many of the geometric and topological mathematical structures of a point-particle kinetic plasma model, including its symplectic structure, symmetries, conservation laws, and cohomology (Villasenor & Buneman 1992;Esirkepov 2001;Squire et al 2012;Evstatiev & Shadwick 2013;Xiao et al 2013;Moon et al 2015;Qin et al 2015;Xiao et al 2015;He et al 2015;Crouseilles et al 2015;Qin et al 2016;He et al 2016;Burby 2017;Morrison 2017;Kraus & Hirvijoki 2017;Xiao et al 2018;Xiao & Qin 2019;Glasser & Qin 2020;Hirvijoki et al 2020;Wang et al 2021-07;Xiao & Qin 2021;Holderied et al 2021;Perse et al 2021;O'Connor et al 2021;Pinto et al 2021). One such structure, gauge symmetry, was first preserved in a PIC code in the Lagrangian formalism via a variational method (Squire et al 2012).…”

Section: Introductionmentioning

confidence: 99%

“…Most of the literature's recent structure-preserving Hamiltonian PIC methods employ a non-canonical Poisson structure to describe particle degrees of freedom (X, V) and discrete electromagnetic fields (E, B) (Qin et al 2015;Xiao et al 2015;He et al 2015;Crouseilles et al 2015;He et al 2016;Burby 2017;Morrison 2017;Kraus & Hirvijoki 2017;Xiao et al 2018;Xiao & Qin 2019Holderied et al 2021;Perse et al 2021;Pinto et al 2021). This approach hides from view the gauge symmetry of the Vlasov-Maxwell system and the simplicity of its canonical Poisson structure, which is characterized by the electromagnetic potential A and its conjugate momentum Y ∼ dA/dt.…”

Section: Introductionmentioning

confidence: 99%

A gauge-compatible Hamiltonian splitting algorithm for particle-in-cell simulations using finite element exterior calculus

Glasser,

Qin

2021

Preprint

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A particle-in-cell algorithm is derived with a canonical Poisson structure in the formalism of finite element exterior calculus. The resulting method belongs to the class of gaugecompatible splitting algorithms, which preserve gauge symmetries and their associated conservation laws to machine precision via the momentum map. With a numerical example of Landau damping, we demonstrate the use of the momentum map to establish initial conditions with a fixed, homogeneous, neutralizing, positive background charge.

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“…The identification of this structure might be important for the construction of structure preserving Hamiltonian algorithms that improve the stability and the fidelity of plasma simulations (Kraus et al 2017;Morrison 2017). Such a structure-preserving code has been recently developed for the simulation of MHD waves that interact with energetic particles in the framework of hybrid kinetic-MHD (Holderied, Possanner & Wang 2021). Moreover, we obtain a second model upon considering the method of Hamiltonian construction of Tronci (2010) for the PCS.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian kinetic-Hall magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes

2021

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We present two generalized hybrid kinetic-Hall magnetohydrodynamics (MHD) models describing the interaction of a two-fluid bulk plasma, which consists of thermal ions and electrons, with energetic, suprathermal ion populations described by Vlasov dynamics. The dynamics of the thermal components are governed by standard fluid equations in the Hall MHD limit with the electron momentum equation providing an Ohm's law with Hall and electron pressure terms involving a gyrotropic electron pressure tensor. The coupling of the bulk, low-energy plasma with the energetic particle dynamics is accomplished through the current density (current coupling scheme; CCS) and the ion pressure tensor appearing in the momentum equation (pressure coupling scheme; PCS) in the first and the second model, respectively. The CCS is a generalization of two well-known models, because in the limit of vanishing energetic and thermal ion densities, we recover the standard Hall MHD and the hybrid kinetic-ions/fluid-electron model, respectively. This provides us with the capability to study in a continuous manner, the global impact of the energetic particles in a regime extending from vanishing to dominant energetic particle densities. The noncanonical Hamiltonian structures of the CCS and PCS, which can be exploited to study equilibrium and stability properties through the energy-Casimir variational principle, are identified. As a first application here, we derive a generalized Hall MHD Grad–Shafranov–Bernoulli system for translationally symmetric equilibria with anisotropic electron pressure and kinetic effects owing to the presence of energetic particles using the PCS.

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MHD-kinetic hybrid code based on structure-preserving finite elements with particles-in-cell

Cited by 20 publications

References 47 publications

A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

A gauge-compatible Hamiltonian splitting algorithm for particle-in-cell simulations using finite element exterior calculus

Hamiltonian kinetic-Hall magnetohydrodynamics with fluid and kinetic ions in the current and pressure coupling schemes

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