Comparison of basis functions defined on infinite interval for pseudo-spectral Vlasov solvers

Abstract:

Vlasov simulation solves the Vlasov equation directly using a mesh in phase space. Although it has a better rate of convergence than popular Particle-in-Cell methods, the use of extra mesh in velocity space requires much more computational resources. Spectral methods are in general superior than finite difference and finite volume methods because they converge much faster and require less grid points for the same target accuracy.

It is well known that properties of a spectral method heavily depend on the choice of basis functions. In this study, three basis sets to expand the distribution function in the velocity space, Fourier series, Hermite functions, and Rational Chebyshev function were compared. Rational Chebyshev functions are Chebyshev polynomials transformed to cover infinite interval and they are known to work well for broad range of problems defined on infinite interval. Hermite functions also have infinite interval but only works well if the solution converges to zero in the same manner as Gaussian distribution function. Fourier series are essentially used to expand periodic functions.

While Fourier spectral method and finite volume types of schemes are adequate to calculate the value of the distribution function at small velocity, Hermite and Rational Chebyshev functions should be better suited when we want to evaluate integrated values such as number density and pressure because they depend on the values of the distribution functions at large velocity as well. This is especially true for distributions with non-thermal components which are very common in space plasmas.

General properties of each basis set and application to standard one-dimensional electrostatic Vlasov simulation test problems such as Landau damping and two-stream instability will be discussed. Fourier and Rational Chebyshev have nearly the same level of accuracy in most cases, whereas Hermite functions only work well when the shape of the distribution function is sufficiently close to Maxwellian throughout the simulation.