Vlasov simulations using velocity-scaled Hermite representations

Journal Article · Mon Aug 10 04:00:00 EDT 1998 · Journal of Computational Physics

DOI:https://doi.org/10.1006/jcph.1998.5925· OSTI ID:653497

Schumer, J W ^[1]; Holloway, J P ^[2]

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1. Naval Research Lab., Washington, DC (United States)
2. Univ. of Michigan, Ann Arbor, MI (United States). Nuclear Engineering and Radiological Sciences

The efficiency, accuracy, and stability of two different pseudo-spectral methods using scaled Hermite basis and weight functions, applied to the nonlinear Vlasov-Poisson equations in one dimension (1d-1v), are explored and compared. A variable velocity scale U is introduced into the Hermite basis and is shown to yield orders of magnitude reduction in errors, as compared to linear kinetic theory, with no increase in workload. A set of Fourier-Hermite coefficients, representing a periodic Gaussian distribution function, are advanced through time with an O({Delta}t{sup 2})-accurate splitting method. Within this splitting scheme, the advection and acceleration terms of the Vlasov equation resolved separately using an O({Delta}t{sup 4})-accurate Runge-Kutta method. The asymmetrically weighted (AW) Hermite basis, which has been used previously by many authors, conserves particles and momentum exactly and total energy to O({Delta}t{sup 3}); however, the AW Hermite method does not conserve the square integral of the distribution and is, in fact, numerically unstable. The symmetrically weighted (SW) Hermite algorithm, applied here to the Vlasov system for the first time, can either conserve particles and energy (for N{sub u} even) or momentum (for N{sub u} odd) as {Delta}t {yields} 0, where N{sub u} is the largest Hermite mode number. The SW Hermite method conserves the square integral of the distribution and, therefore, remains numerically stable. In addition, careful velocity scaling improves the conservation properties of the SW Hermite method. Damping and growth rates, oscillation frequencies, E-field saturation levels, and phase-space evolution are seen to be qualitatively correct during simulations. Relative errors with respect to linear Landau damping and linear bump-on-tail instability are shown to be less than 1% using only 64 velocity-scaled Hermite functions.

• Sponsoring Organization: National Science Foundation, Washington, DC (United States); USDOE, Washington, DC (United States)
• OSTI ID: 653497
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 2 Vol. 144; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

70 PLASMA PHYSICS AND FUSION TECHNOLOGY
99 GENERAL AND MISCELLANEOUS
ALGORITHMS
BOLTZMANN-VLASOV EQUATION
BUMP-IN-TAIL INSTABILITY
CALCULATION METHODS
COMPARATIVE EVALUATIONS
HERMITE POLYNOMIALS
PLASMA SIMULATION
RUNGE-KUTTA METHOD
WEIGHTING FUNCTIONS