Numerical Integration of Kinetic Equations

Dolinsky, A. A.

doi:10.1063/1.1761243

Cited by 19 publications

(7 citation statements)

References 6 publications

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“…We find, however, that there is not much difference between calculating the distributions with the full Lenard-Balescu collision operator versus using Landau with a well-chosen Coulomb logarithm, even in cases where the temperature separation is large. This is consistent with previous numerical comparisons of the two equations [5,6].…”

Section: Introductionsupporting

confidence: 93%

“…For the dielectric function we use the classical approximation; that is, we neglect quantum diffraction [5]. The main benefit of this is that the dielectric function takes the comparatively simple form (6). That is, the variables X and Y separate, facilitating analytical evaluation of the X-integral without losing any important physical content.…”

Section: Equation Coefficientsmentioning

confidence: 99%

“…In addition to the Landau equation, our method can be adapted to solve a range of other kinetic equations, and we demonstrate this here by solving the non-degenerate quantum Lenard-Balescu equation. This equation has rarely been attempted numerically [6,5] because it requires an integration in frequency and wavenumber over a dielectric function. This dielectric function adds another degree of non-linearity to the system because it contains the distribution function, although this is not the main source of difficulty.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive spectral solution method for the Landau and Lenard-Balescu equations

Scullard

Hickok

Sotiris

et al. 2020

Journal of Computational Physics

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We present an adaptive spectral method for solving the Landau/Fokker-Planck equation for electron-ion systems. The heart of the algorithm is an expansion in Laguerre polynomials, which has several advantages, including automatic conservation of both energy and particles without the need for any special discretization or time-stepping schemes. One drawback is the O(N 3 ) memory requirement, where N is the number of polynomials used. This can impose an inconvenient limit in cases of practical interest, such as when two particle species have widely separated temperatures. The algorithm we describe here addresses this problem by periodically re-projecting the solution onto a judicious choice of new basis functions that are still Laguerre polynomials but have arguments adapted to the current physical conditions. This results in a reduction in the number of polynomials needed, at the expense of increased solution time. Because the equations are solved with little difficulty, this added time is not of much concern compared to the savings in memory. To demonstrate the algorithm, we solve several relaxation problems that could not be computed with the spectral method without re-projection. Another major advantage of this method is that it can be used for collision operators more complicated than that of the Landau equation, and we demonstrate this here by using it to solve the non-degenerate quantum Lenard-Balescu equation for a hydrogen plasma.

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Section: Introductionsupporting

confidence: 93%

Section: Equation Coefficientsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive spectral solution method for the Landau and Lenard-Balescu equations

Scullard

Hickok

Sotiris

et al. 2020

Journal of Computational Physics

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“…These difficulties, coupled with the fact that the Landau equation, despite its deficiencies, yields distributions that are likely qualitatively correct at weak coupling, have kept the Lenard-Balescu equation from being studied numerically in any serious way. We know of only one previous attempt: Dolinsky's pioneering 1965 solution of the classical LB equation 14 using a discretization method in velocity. This work predates the advent of conservative velocity discretization schemes even for the Fokker-Planck equation, but it is not completely clear that such methods are generically well-suited to the Lenard-Balescu equation anway because of the need to accurately integrate over the features of the dielectric function.…”

Section: Quantum Lenard-balescu Equationmentioning

confidence: 99%

Numerical solution of the quantum Lenard-Balescu equation for a non-degenerate one-component plasma

et al. 2016

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We present a numerical solution of the quantum Lenard-Balescu equation using a spectral method, namely an expansion in Laguerre polynomials. This method exactly conserves both particles and energy and facilitates the integration over the dielectric function. To demonstrate the method, we solve the equilibration problem for a spatially homogeneous one-component plasma with various initial conditions. Unlike the more usual Landau/Fokker-Planck system, this method requires no input Coulomb logarithm; the logarithmic terms in the collision integral arise naturally from the equation along with the non-logarithmic order-unity terms. The spectral method can also be used to solve the Landau equation and a quantum version of the Landau equation in which the integration over the wavenumber requires only a lower cutoff. We solve these problems as well and compare them with the full Lenard-Balescu solution in the weak-coupling limit. Finally, we discuss the possible generalization of this method to include spatial inhomogeneity and velocity anisotropy.

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“…Eq. (42) may also be integrated using a non-conservative method [30,16], but when this was tried in our code the distribution function, even when initialized as a Maxwellian, was visibly modified by numerical effects within a few collision times. This can be mitigated by using ultra-fine resolution in momentum-space, but such a grid would require a very small time-step and an enormous amount of computer memory.…”

Section: Characteristic Collisional Quantitiesmentioning

confidence: 99%