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

Journal Article · Tue Nov 12 00:00:00 EST 2019 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2019.109110· OSTI ID:1670540

Scullard, Christian R. ^[1]; Hickok, Abigail ^[2]; Sotiris, Justyna O. ^[3]; Tzolova, Bilyana M. ^[4]; Van Heyningen, R. Loek ^[5]; Graziani, Frank R. ^[1]

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1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2. Univ. of California, Los Angeles, CA (United States)
3. Univ. of California, Los Angeles, CA (United States); Univ. of New Mexico, Albuquerque, NM (United States)
4. Univ. of California, Los Angeles, CA (United States); Johns Hopkins Univ., Baltimore, MD (United States)
5. Univ. of California, Los Angeles, CA (United States); Univ. of California, Berkeley, CA (United States)

In this paper, 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 of such an expansion is the O(N³) 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 (QLB) equation for a hydrogen plasma. We conclude with some comparisons of temperature relaxation problems solved with the latter equation and the Landau equation with a Coulomb logarithm inspired by the properties of the QLB operator. We find that with this choice of Coulomb logarithm, there is little difference between using the two equations for these particular systems.

• Research Organization: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC52-07NA27344
• OSTI ID: 1670540
• Alternate ID(s): OSTI ID: 1577931
• Report Number(s): LLNL-JRNL--766988; 953984
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: na Vol. 402; ISSN 0021-9991
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

70 PLASMA PHYSICS AND FUSION TECHNOLOGY
Kinetic theory
Landau equation
Lenard-Balescu equation