A numerical method for solving the Vlasov–Poisson equation based on the conservative IDO scheme

Imadera, Kenji; Kishimoto, Y.; Saito, Daisuke N.; Li, Jiquan; Utsumi, Takayuki

doi:10.1016/j.jcp.2009.09.008

Cited by 14 publications

(15 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, the matrix L 1 corresponds to the linear advection operator on the right hand side of Eq. (15), while N is the non-linear operator associated with the convolutions. Finally, the matrices L 2 and L 3 are defined by the right hand side of Eqs.…”

Section: The Methodsmentioning

confidence: 99%

“…(15) and (22) that ΔC 0,0,0,s 0,0,0 /Δt = 0 and therefore the total mass in the system is conserved.…”

mentioning

confidence: 94%

“…For spectral methods, this is brought by the fact that one might need a large number of moments to capture the details of the distribution function. However, the continuous advance in computer power mitigates these limitations and continues to drive research in Eulerian Vlasov [12][13][14][15][16][17] and spectral methods [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Delzanno

2015

Journal of Computational Physics

View full text Add to dashboard Cite

A spectral method for the numerical solution of the multi-dimensional VlasovMaxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank-Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian-Free Newton-Krylov method.It is shown analytically that the Fourier-Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

show abstract

Section: The Methodsmentioning

confidence: 99%

“…(15) and (22) that ΔC 0,0,0,s 0,0,0 /Δt = 0 and therefore the total mass in the system is conserved.…”

mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Delzanno

2015

Journal of Computational Physics

View full text Add to dashboard Cite

show abstract

“…To perform kinetic simulations with non-periodic boundaries, a conservative Vlasov-Maxwell scheme based on the finite-difference manner is required. In one work, the Vlasov equations of the conservative form were discretized with the conservative form of the interpolated differential operator (IDO-CF) method [14], but the errors of energy conservation were much larger than the round-off level [15]. In gyrokinetic simulations, there is a charge-conserving algorithm based on finite-difference methods, although the momentum and energy cannot be conserved [16,17,18].…”

mentioning

confidence: 99%

Quadratic conservative scheme for relativistic Vlasov–Maxwell system

Shiroto

Ohnishi

Sentoku

2019

Journal of Computational Physics

View full text Add to dashboard Cite

For more than half a century, most of the plasma scientists have encountered a violation of the conservation laws of charge, momentum, and energy whenever they have numerically solve the first-principle equations of kinetic plasmas, such as the relativistic Vlasov-Maxwell system. This fatal problem is brought by the fact that both the Vlasov and Maxwell equations are indirectly associated with the conservation laws by means of some mathematical manipulations. Here we propose a quadratic conservative scheme, which can strictly maintain the conservation laws by discretizing the relativistic Vlasov-Maxwell system. A discrete product rule and summation-by-parts are the key players in the construction of the quadratic conservative scheme. Numerical experiments of the relativistic two-stream instability and relativistic Weibel instability prove the validity of our computational theory, and the proposed strategy will open the doors to the first-principle studies of mesoscopic and macroscopic plasma physics.where ρ and J are the charge and current densities, respectively. However, Eqs. (5) and (6) are naturally satisfied when the law of charge conservation and the inexistence of the magnetic monopole are assumed. Fortunately, these principles are derived from the 0th-order moment equation of Eq.(1). Therefore, we do not need to solve Eqs. (5) and (6) when solving Eqs. (1), (3), and (4). Modern numerical investigations of kinetic plasmas can be characterized in two ways. The first one is a particle-in-cell (PIC) method [1], which solves the equations of motion of charged particles, such as ions and electrons, instead of the Vlasov equation. In the PIC, the equations of motion are coupled with the Maxwell equations using some of the field interpolation techniques. Another approach is to discretize the Vlasov equations directly by using the finite-difference method, spectral method, and so on (hereafter, called "Vlasov simulation"). However, these numerical methods have a fatal problem; the conservation laws of charge, momentum, and energy are violated in principle when the governing equations are discretized. The term "numerical heating" is a nightmare among PIC users, which implies that the total energies in PIC simulations increase infinitely even if there is no physical energy source. To overcome this issue, many mathematical investigations were performed on the conservation property of the first-principle kinetic simulations, and significant progress was made mainly in the 2010s. Crank-Nicolson time integration is one of the key structures in the construction of conservative PIC methods; recent studies have employed it in energyconserving [2, 3, 4], charge-energy-conserving one-dimensional one-momentum-component (1D1P) [5], onedimensional three-momentum-components (1D3P) [6], and two-dimensional three-momentum-component (2D3P) [7] PIC methods. A study discretized the equations of motion with a leap-frog method, while the Maxwell equations were solved with the Crank-Nicolson method; the total energy was strictly conserv...

show abstract

“…(4), which corresponds to the source term driving the spatial dynamics of s (1) and s (2) . In this study, we investigate the entropy dynamics in ITG turbulence with global profile relaxation using a 4D (i.e., 3D in real space and 1D in velocity space) gyrokinetic full-f Vlasov simulation based on the IDO-CF scheme [6]. We employ a shear-less slab geometry with a system size of L x = 2L y = 64 and L z = 8000 in real space and L v = 10 in velocity space.…”

mentioning

confidence: 99%

Global Profile Relaxation and Entropy Dynamics in Turbulent Transport

Imadera

Kishimoto

2010

Plasma and Fusion Research

Self Cite

View full text Add to dashboard Cite

A hierarchical entropy balance equation retaining the dynamics in the radial direction is introduced to study non-local turbulent transport and the associated global profile relaxation. It consists of first-and second-order equations that describe the entropy dynamics related to thermodynamics/fluid quantity and the corresponding micro-scale phase space fluctuations, respectively. Specifically, the second-order equation describes not only a local entropy production related to heat and density flux (i.e., zonal flow), but also the spatial convection of perturbed entropy. We investigated the entropy dynamics in ion-temperature-gradient driven turbulence based on a global gyrokinetic Vlasov simulation in slab geometry. Entropy convection plays an important role in the relaxation dynamics dominated by the avalanche process. A self-organized relaxed state is established, in which short-wavelength temperature corrugation, i.e., zonal pressure, is regulated by zonal flow shear. Turbulent transport in magnetically confined fusion plasmas exhibits various prominent features characterized by different temporal and spatial scales. Zonal modes, such as zonal flow and pressure, which are poloidally and toroidally symmetric macro-scale structures nonlinearly generated from micro-scale turbulence, play an important role in regulating turbulent structure and then transport [1]. These zonal modes are considered to be tightly linked to non-local characteristics of turbulent transport that are not explained by the Gaussian statistics, such as intermittent transport, turbulent spreading [2], and avalanche dynamics, etc. Self-organized critical (SOC) transport [3] is also an example in which the turbulence provides a strong constraint on relaxation, leading to a self-organized stiff temperature profile.To characterize such transport dynamics, phase space entropy, which connects micro-scale turbulent structure to macro-scale thermodynamic quantities, has been introduced [4]. However, the entropy has been generally treated as a global quantity that is integrated over phase space, so that the effects of zonal flow and turbulent spreading do not appear explicitly in the entropy balance equation. As the result, the non-local nature of transport is hardly discussed.To address this problem, we extend the entropy balance equation by retaining the dynamics in the radial direction. We start with a four-dimensional (4D) gyrokinetic model for electrostatic ion-temperature-gradient (ITG) driven turbulence in slab geometry. The normalized basic equation system is given by the gyrokinetic Vlasov

show abstract

A numerical method for solving the Vlasov–Poisson equation based on the conservative IDO scheme

Cited by 14 publications

References 34 publications

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Multi-dimensional, fully-implicit, spectral method for the Vlasov–Maxwell equations with exact conservation laws in discrete form

Quadratic conservative scheme for relativistic Vlasov–Maxwell system

Global Profile Relaxation and Entropy Dynamics in Turbulent Transport

Contact Info

Product

Resources

About