Abstract Details
Abstracts
Author: Tonatiuh Sanchez-Vizuet
Requested Type: Poster
Submitted: 2017-03-16 13:44:02
Co-authors: A. Cefon, J. Wilkening
Contact Info:
Courant Institute of Mathematical Sciences (NYU)
Warren Weaver Hall. 251 Mercer
New York, New York 10012-1
United States
Abstract Text:
We explore the approximation and stability properties of a new discretization strategy for kinetic equations with energy diffusion due to Fokker-Planck collisions. The discretization scheme is based on pseudospectral collocation on a grid defined by the zeros of a non-standard family of orthogonal polynomials.
The class of polynomials in question, hereby referred to as Maxwell
polynomials, is orthogonal with respect to the inner product defined by the integration over the positive half real line with weight function $w(x)= x^2 exp^{-x^2}$. The remarkable approximation and interpolation properties of these polynomials for functions with Maxwellian behavior had been demonstrated previously for steady state problems [2,3] and for time-dependent diffusion equations [4]. Importantly, these studies did not consider pseudospectral collocation implementations, even though collocation is often favored for numerical solvers of kinetic equations [1].
The pseudospectral approximation of differential operators results in non-normal matrices whose stability properties are not completely
determined by their spectrum. We discuss the role of the
(epsilon-)peudospectrum of the discrete operator and its sensitivity to reformulations of the continuous equation in connection with the time stability of the fully discrete system. We propose optimized formulations for the time stepping schemes most commonly used in the plasma physics community, and compare our results with those obtained with the discretization schemes of popular gyrokinetic codes.
Partially funded by the US Department of Energy. Grant No. DE-FG02-86ER53233.
[1] J. Candy and E.A. Belli and R.V. Bravenec. JCP, 324, 73 (2016).
[2] M. Landreman and D. Ernst JCP, 243, 130 (2013).
[3] "Spectral Methods in Chemistry and Physics" B. Shizgal. Springer, 2015.
[4] J. Wilkening and A. Cerfon, and M. Landreman. JCP 294,58 (2015).
Comments: