Relativistic Vlasov-Maxwell modelling using finite volumes and adaptive mesh refinement

Wettervik, Benjamin Svedung; DuBois, Timothy C.; Siminos, Evangelos; Fülöp, Tünde

doi:10.1140/epjd/e2017-80102-2

Cited by 9 publications

(12 citation statements)

References 58 publications

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“… Adaptively refined grids can be employed to reduce resolution and thus computational expense in areas of phase space that are considered less important for the physical problem at hand (Wettervik et al. 2017 ; Besse et al. 2008 ).…”

Section: Numerical Modelling and Hpc Aspectsmentioning

confidence: 99%

Vlasov methods in space physics and astrophysics

Palmroth

Ganse

Pfau‐Kempf

et al. 2018

Living Rev Comput Astrophys

140

143

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This paper reviews Vlasov-based numerical methods used to model plasma in space physics and astrophysics. Plasma consists of collectively behaving charged particles that form the major part of baryonic matter in the Universe. Many concepts ranging from our own planetary environment to the Solar system and beyond can be understood in terms of kinetic plasma physics, represented by the Vlasov equation. We introduce the physical basis for the Vlasov system, and then outline the associated numerical methods that are typically used. A particular application of the Vlasov system is Vlasiator, the world’s first global hybrid-Vlasov simulation for the Earth’s magnetic domain, the magnetosphere. We introduce the design strategies for Vlasiator and outline its numerical concepts ranging from solvers to coupling schemes. We review Vlasiator’s parallelisation methods and introduce the used high-performance computing (HPC) techniques. A short review of verification, validation and physical results is included. The purpose of the paper is to present the Vlasov system and introduce an example implementation, and to illustrate that even with massive computational challenges, an accurate description of physics can be rewarding in itself and significantly advance our understanding. Upcoming supercomputing resources are making similar efforts feasible in other fields as well, making our design options relevant for others facing similar challenges.

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Section: Numerical Modelling and Hpc Aspectsmentioning

confidence: 99%

Vlasov methods in space physics and astrophysics

Palmroth

Ganse

Pfau‐Kempf

et al. 2018

Living Rev Comput Astrophys

140

143

View full text Add to dashboard Cite

show abstract

“…Such flexibility in velocity space grids is not guaranteed to provide significant advantage over uniform, fixed, grids, as processes such as phase-mixing can create structure everywhere in velocity space at arbitrarily small velocity space scales; however, for simulations which include significant particle acceleration such as the electrostatic shock in Section 5.2.3, the generation of the highly energetic tail in the proton distribution function would be much cheaper. And relativistic laserplasma Vlasov codes in 1X1P have demonstrated significant gains with AMR [15]. Most importantly, as discussed in Sections 5.2.1 and 5.3.2, there is currently nothing inhibiting the distribution function from going negative, nor is there anything preventing the equation system from accumulating errors in the divergence relations in Maxwell's equations, Eqns.…”

Section: Summary and Future Workmentioning

confidence: 99%

“…While computationally infeasible in the past due to the need to solve a six-dimensional system (three position and three velocity), plus time, to accurately capture the dynamics of the plasma, in recent years, directly solving the Vlasov equation has become a more popular alternative approach to illuminating the microphysics of the plasma's evolution. Previous Vlasov work has focused almost exclusively on the so-called hybrid framework, treating the electrons as a fluid, usually massless and isothermal, to save significantly on computational cost [6,7,8,9,10,11,12,13,14], with some exceptions [15,16]. In this paper, we present the addition of a multi-species Vlasov-Maxwell solver to Gkeyll, a modular framework in which a variety of plasma physics and fluid dynamics solvers are currently being built [17,18,19,20,21,22,23,24].…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin algorithms for fully kinetic plasmas

Juno¹,

Hakim²,

TenBarge³

et al. 2018

Journal of Computational Physics

114

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We present a new algorithm for the discretization of the Vlasov-Maxwell system of equations for the study of plasmas in the kinetic regime. Using the discontinuous Galerkin finite element method for the spatial discretization, we obtain a high order accurate solution for the plasma's distribution function. Time stepping for the distribution function is done explicitly with a third order strong-stability preserving Runge-Kutta method. Since the Vlasov equation in the Vlasov-Maxwell system is a high dimensional transport equation, up to six dimensions plus time, we take special care to note various features we have implemented to reduce the cost while maintaining the integrity of the solution, including the use of a reduced high-order basis set. A series of benchmarks, from simple wave and shock calculations, to a five dimensional turbulence simulation, are presented to verify the efficacy of our set of numerical methods, as well as demonstrate the power of the implemented features

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“…Numerical integrators based on fully discretizing phase space, i.e. the Eulerian approach, have become affordable only in recent years, both due to improved algorithms and the greatly increased performance of computer systems (see, e.g., [5,33,13,12,17,39,35,37]). Algorithms based on splitting methods have been considered.…”

Section: Introductionmentioning

confidence: 99%

A low-rank projector-splitting integrator for the Vlasov–Maxwell equations with divergence correction

Einkemmer

Ostermann

Piazzola

2020

Journal of Computational Physics

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The Vlasov-Maxwell equations are used for the kinetic description of magnetized plasmas. As they are posed in an up to 3+3 dimensional phase space, solving this problem is extremely expensive from a computational point of view. In this paper, we exploit the low-rank structure in the solution of the Vlasov equation. More specifically, we consider the Vlasov-Maxwell system and propose a dynamic low-rank integrator. The key idea is to approximate the dynamics of the system by constraining it to a low-rank manifold. This is accomplished by a projection onto the tangent space. There, the dynamics is represented by the low-rank factors, which are determined by solving lower-dimensional partial differential equations. The proposed scheme performs well in numerical experiments and succeeds in capturing the main features of the plasma dynamics. We demonstrate this good behavior for a range of test problems. The coupling of the Vlasov equation with the Maxwell system, however, introduces additional challenges. In particular, the divergence of the electric field resulting from Maxwell's equations is not consistent with the charge density computed from the Vlasov equation. We propose a correction based on Lagrange multipliers which enforces Gauss' law up to machine precision.

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Relativistic Vlasov-Maxwell modelling using finite volumes and adaptive mesh refinement

Cited by 9 publications

References 58 publications

Vlasov methods in space physics and astrophysics

Vlasov methods in space physics and astrophysics

Discontinuous Galerkin algorithms for fully kinetic plasmas

A low-rank projector-splitting integrator for the Vlasov–Maxwell equations with divergence correction

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