In
this p
aper, we propose
a high order conserv
ative se
mi-L
agr
angi
an (SL) fini
te difference Her
mi
te weigh
ted essen
ti
ally non-oscill
atory (HWENO)
me
thod for
the Vl
asov equ
ation b
ased on di
mension
al spli
tting. HWENO w
as firs
t proposed for solving nonline
ar hyperbolic proble
ms by evolving bo
th func
tion v
alues
and i
ts firs
t deriv
ative v
alues (Qiu
and Shu (2004)
an id="bbr0230">[23]a>an>). The
major
adv
an
tage of HWENO, co
mp
ared wi
th
the origin
al WENO, lies in i
ts co
mp
ac
tness in recons
truc
tion s
tencils.
There are several new ingredients in this paper. Firstly we propose a m>mass-conservativem> SL HWENO scheme for a 1-D equation by working with a flux-difference form, following the work of Qiu and Christlieb (2010) an id="bbr0250">[25]a>an>. Secondly, we propose a proper splitting for equations of partial derivatives in HWENO framework to ensure local mass conservation. The proposed fifth order SL HWENO scheme with dimensional splitting has been tested to work well in capturing filamentation structures without oscillations when the time step size is within the Eulerian CFL constraint. However, when the time stepping size becomes larger, numerical oscillations are observed for the ‘mass conservative’ dimensional splitting HWENO scheme, as there are extra source terms in equations of partial derivatives. In this case, we introduce WENO limiters to control oscillations. Classical numerical examples on linear passive transport problems, as well as the nonlinear Vlasov–Poisson system, have been tested to demonstrate the performance of the proposed scheme.