Conservative fourth-order finite-volume Vlasov–Poisson solver for axisymmetric plasmas in cylindrical (r,v,v) phase space coordinates

Vogman, G. V.; Shumlak, U.; Colella, Phillip

doi:10.1016/j.jcp.2018.07.029

Cited by 25 publications

(3 citation statements)

References 82 publications

(122 reference statements)

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“…In [17], this code was extended to mapped multi block grids and tested on a sinusoidally distorted mesh with advection problems. Vogman et al [22] introduced a continuum code with finite volume discretisation for an electrostatic axisymmetric cylindrical Vlasov-Poisson system using specular reflection as particle boundary conditions and Dirichlet boundary conditions for the fields.…”

Section: Related Workmentioning

confidence: 99%

Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates

Perse¹,

Kormann²,

Sonnendrücker³

2021

SIAM J. Sci. Comput.

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Structure-preserving methods can be derived for the Vlasov-Maxwell system from a discretisation of the Poisson bracket with compatible finite-elements for the fields and a particle representation of the distribution function. These geometric electromagnetic particle-in-cell (GEMPIC) discretisations feature excellent conservation properties and long-time numerical stability. This paper extends the GEM-PIC formulation in curvilinear coordinates to realistic boundary conditions. We build a de Rham sequence based on spline functions with clamped boundaries and apply perfect conductor boundary conditions for the fields and reflecting boundary conditions for the particles. The spatial semi-discretisation forms a discrete Poisson system. Time discretisation is either done by Hamiltonian splitting-yielding a semi-explicit Gauss conserving scheme-or by a discrete gradient scheme applied to a Poisson splitting-yielding a semi-implicit energy-conserving scheme. Our system requires the inversion of the spline finite element mass matrices, which we precondition with the combination of a Jacobi preconditioner and the spectrum of the mass matrices on a periodic tensor product grid. * Funding: This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. B. P. has also been supported by Deutsche Forschungsgemeinschaft (DFG) through the TUM International Graduate School of Science and Engineering (IGSSE).

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Section: Related Workmentioning

confidence: 99%

Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates

Perse¹,

Kormann²,

Sonnendrücker³

2021

SIAM J. Sci. Comput.

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“…We note that the symmetry boundary condition -identical to the specular reflection boundary condition considered in Ref. [63]-is a special case of the treatment above in the limit of vanishing wall velocity and is used for the r = 0 boundary. An illustration of the elastic moving wall boundary condition is shown in Fig.…”

Section: Elastic Moving Wall and Symmetry Boundary Conditionsmentioning

confidence: 99%

An Eulerian Vlasov-Fokker-Planck Algorithm for Spherical Implosion Simulations of Inertial Confinement Fusion Capsules

Taitano,

Keenan,

Chacon

et al. 2020

Preprint

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We present a numerical algorithm that enables a phase-space adaptive Eulerian Vlasov-Fokker-Planck (VFP) simulation of an inertial confinement fusion (ICF) capsule implosion. The approach relies on extending a recent mass, momentum, and energy conserving phase-space moving-mesh adaptivity strategy to spherical geometry. In configuration space, we employ a mesh motion partial differential equation (MMPDE) strategy while, in velocity space, the mesh is expanded/contracted and shifted with the plasma's evolving temperature and drift velocity. The mesh motion is dealt with by transforming the underlying VFP equations into a computational (logical) coordinate, with the resulting inertial terms carefully discretized to ensure conservation. To deal with the spatial and temporally varying dynamics in a spherically imploding system, we have developed a novel nonlinear stabilization strategy for MMPDE in the configuration space. The strategy relies on a nonlinear optimization procedure that optimizes between mesh quality and the volumetric rate change of the mesh to ensure both accuracy and stability of the solution. Implosions of ICF capsules are driven by several boundary conditions: 1) an elastic moving wall boundary; 2) a time-dependent Maxwellian Dirichlet boundary; and 3) a pressure-driven Lagrangian boundary. Of these, the pressure-driven Lagrangian boundary driver is new to our knowledge. The implementation of our strategy is verified through a set of test problems, including the Guderley and Van-Dyke implosion problems -the first-ever reported using a Vlasov-Fokker-Planck model.

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“…However, when it is not possible to keep them all, they can be used to monitor the validity of the simulation by checking accuracy of these invariants. Many attempts have been made for solving Vlasov-Poisson system, including classical discretizations as finite difference methods [2], finite element method [60,47,1], finite volume method [32,33,18,54] , spectral method [42], discontinuous Galerkin methods [37,22,12,11,59,50,39], statistical based method as particle-in-cell method [14,53,24,34,23,19]. There is also another important category named Semi-Lagrangian methods [46,4,20,44,43,52,9,55,5,30,31], which is popular thanks to its good precision and as it is free from time step limitation.…”

Section: Introductionmentioning

confidence: 99%

Highly accurate monotonicity-preserving Semi-Lagrangian scheme for Vlasov-Poisson simulations

Yang

Mehrenberger

2021

Journal of Computational Physics

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In this paper, we study a highly accurate monotonicity-preserving (MP) Semi-Lagrangian scheme for Vlasov-Poisson simulations. The classical Semi-Lagrangian scheme is known to be highly accurate and free from CFL condition, but it does not satisfy local maximum principle. To remedy this drawback, using the conservative form of the Semi-Lagrangian scheme, we recast existing MP schemes for the numerical flux in a common framework, and then substitute the local extremum by some "better" guess, in order to avoid as much as possible loss of accuracy and clipping near extrema, while keeping the monotonicity on monotone portions. With the limiter, on the one hand, the scheme keeps the good properties of the unlimited scheme: it is conservative, free from CFL condition and highly accurate. On the other hand, for locally monotonic data, the monotonicity of the solution is preserved. Numerical tests are made on free transport equation and Vlasov-Poisson system illustrating that the limited scheme is more diffusive compared to cubic splines but has better L 1 conservation, which is primarily an advantage for problems with sharp gradients.

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Conservative fourth-order finite-volume Vlasov–Poisson solver for axisymmetric plasmas in cylindrical (r,v,v) phase space coordinates

Cited by 25 publications

References 82 publications

Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates

Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates

An Eulerian Vlasov-Fokker-Planck Algorithm for Spherical Implosion Simulations of Inertial Confinement Fusion Capsules

Highly accurate monotonicity-preserving Semi-Lagrangian scheme for Vlasov-Poisson simulations

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