Fine‐Scale Structures and Negative‐Density Regions: Comparison of Numerical Methods for Solving the Advection Equation

Abstract

A common feature of the Vlasov equation is that it develops fine‐scale filamentation as time evolves, as observed, for example, in global nonlinear simulations of the ion‐temperature‐gradient instability. From a numerical point of view, it is not trivial to simulate nonlinear regimes characterized by increasingly smaller scales, which eventually become smaller than the (finite) grid size. When very small structures occur, higher order interpolation schemes have a tendency to produce overshoots and negative‐density regions unless some additional dissipative procedure is applied. Different interpolation schemes for the distribution function are compared and discussed.

Keywords:

• Advection equation
• interpolation methods
• semi‐Lagrangian

Acknowledgments

We thank G. Manfredi and R.G.L. Vann for useful discussions. This work was partly supported by the Swiss National Science Foundation.

Notes

¹In normalized units, time is normalized to the inverse plasma frequency ω _p ⁻¹, space to the Debye length λ _D , and velocity to the thermal speed V _T =λ _D ω _p .

²Note that these slope correctors do not preserve the monoticity of f. For more details, see Arber and Vann (2002).

³In the case of CS, entropy is calculated as S=−∫| f | ln | f | dx dv.