Vlasov Code Simulations
*This work has been supported by
GrantInAid (KAKENHI) for Young Scientists (Start Up) No.19840024 "Study of revolutionary Vlasov simulation techniques"
GrantInAid (KAKENHI) for Young Scientists (B) No.21740352 "Study of Vlasov simulation techniques for nextgeneration supercomputers"
GrantInAid (KAKENHI) for Young Scientists (B) No.23740367 "Study of crossscale coupling in space plasma via largescale Vlasov simulations"
GrantInAid (KAKENHI) for Challenging Exploratory Research No.25610144 "Sixdimensional Vlasov simulation"
Numerical Code for Educational Purpose
The numerical code has been developed for the 11th International School / Symposium for Space Simulations (ISSS11) and modified for 13th International School / Symposium for Space Simulations (ISSS13).
Application Examples of Multidimensional simulations
Various physical processes in space plasma have been simulated
with the massivelyparallel multidimensional Vlasov code.
 T. Umeda, Simulation of collisionless plasma with the Vlasov method, In: Computer Physics, edited by B. S. Doherty and A. N. Molloy, pp.315332, Nova ScienceNew York, 2012.
 T. Umeda, K. Fukazawa, Y. Nariyuki, and T. Ogino, A scalable full electromagnetic Vlasov solver for crossscale coupling in space plasma, IEEE Transactions on Plasma Science, Vol.40, 14211428, 2012.
Magnetic Reconnection
 T. Umeda, K. Togano, and T. Ogino,
Twodimensional fullelectromagnetic Vlasov code with conservative scheme
and its application to magnetic reconnection,
Comput. Phys. Commun., Vol.180, 365374, 2009.
 T. Umeda, K. Togano, and T. Ogino,
Structures of diffusion regions in collisionless magnetic reconnection,
Phys. Plasmas, Vol.17, 052103, 2010.
 S. Zenitani and T. Umeda, Some remarks on the diffusion regions in magnetic reconnection, Phys. Plasmas, Vol.21, 034503, 2014.
KelvinHelmholtz Instability
 T. Umeda, J. Miwa, Y. Matsumoto, T. K. M. Nakamura, K. Togano,
K. Fukazawa, and I. Shinohara, Full electromagnetic Vlasov code simulation
of the KelvinHelmholtz instability, Phys. Plasmas, Vol.17, 052311, 2010.
 T. Umeda, S. Ueno, and T. K. M. Nakamura, Ion kinetic effects to nonlinear processes of the KelvinHelmholtz instability, Plasma Phys. Contr. Fusion, Vol.56, 075006, 2014.
RayleighTaylor instability
 T. Umeda and Y. Wada, Secondary instabilities in the collisionless RayleighTaylor instability: Full kinetic simulation, Phys. Plasmas, Vol.23, 112117, 2016.
 T. Umeda and Y. Wada, NonMHD effects in the nonlinear development of the MHDscale RayleighTaylor instability, Phys. Plasmas, Vol.24, 072307, 2017.
Interaction between plasma wind and bodies (planets/satellites)
 T. Umeda, T. Kimura, K. Togano, K. Fukazawa, Y. Matsumoto,
T. Miyoshi, N. Terada, T. K. M. Nakamura, and T. Ogino,
Vlasov simulation on the interaction between solar wind
and a dielectric body, Phys. Plasmas, Vol.18, 012908, 2011.
 T. Umeda, Effect of ion cyclotron motion on the structure of wakes: A Vlasov simulation, Earth Planets Space, Vol.64, 231236, 2012.
 T. Umeda and Y. Ito, Entry of solarwind ions into the wake of a small body with a magnetic anomaly: A global Vlasov simulation, Planet. Space Sci., Vol.9394, 3540, 2014.
 T. Umeda and K. Fukazawa, A highresolution global Vlasov simulation of a small dielectric body with a weak intrinsic magnetic field on the K computer, Earth Planets Space, Vol.67, 49, 2015.
Twotrilliongrids simulation on the K computer
Numerical Schemes
Vlasov equation
The Vlasov equation
\(
\frac{\partial f_s}{\partial t} + \mathbf{v}\cdot \frac{\partial f_s}{\partial \mathbf{x}} + \frac{q_s}{m_s}\left( \mathbf{E} + \mathbf{v} \times \mathbf{B}
\right) \cdot \frac{\partial f_s}{\partial \mathbf{v}} = 0
\)
is separated into the following three subequations based on the operator splitting.
\(
\frac{\partial f_s}{\partial t} + \mathbf{v}\cdot \frac{\partial f_s}{\partial \mathbf{x}} = 0
\)
\(
\frac{\partial f_s}{\partial t}
+ \frac{q_s}{m_s}\mathbf{E} \cdot \frac{\partial f_s}{\partial \mathbf{v}} = 0
\)
\(
\frac{\partial f_s}{\partial t} + \frac{q_s}{m_s}\left( \mathbf{v} \times \mathbf{B}
\right) \cdot \frac{\partial f_s}{\partial \mathbf{v}} = 0
\)
The first two advection equations are solved with a multidimensional unsplit advection method, while the third rotation equation is solved with the backsubstitution method. For solving these equations, a highorder conservative and nonoscillatory scheme is used, which exactly satisfies the continuity equation for charge
\(
\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0
\).
 T. Umeda, K. Togano, and T. Ogino,
Twodimensional fullelectromagnetic Vlasov code with conservative scheme
and its application to magnetic reconnection,
Comput. Phys. Commun., Vol.180, 365374, 2009.
 H. Schmitz and R. Grauer,
Comparison of time splitting and backsubstitution methods
for integrating Vlasov's equation with magnetic fields,
Comput. Phys. Commun., Vol.175, 8692, 2006.
 T. Umeda,
A conservative and nonoscillatory scheme for Vlasov code simulations,
Earth Planets Space, Vol.60, 773779, 2008.
 T. Umeda, Y. Nariyuki, and D. Kariya,
A nonoscillatory and conservative semiLagrangian scheme
with fourthdegree polynomial interpolation for solving the Vlasov equation,
Comput. Phys. Commun., Vol.183, 10941100, 2012.
Maxwell equations
The Maxwell equations are solved with the FiniteDifference TimeDomain (FDTD) method based on the Yee grid system, as standard particleincell codes do.
 K.S. Yee,
Numerical solution of initial boundary value problems
involving Maxwell's equations in isotropic media,
IEEE Trans. Antennas Propag., Vol.AP14, 302307, 1966.
