  The term `Convected Scheme' (CS) refers to a family of algorithms, most
usually applied to the solution of Boltzmann's equation, which uses a method of
characteristics in an integral form to project an initial cell forward to a
group of final cells. As such the CS is a `forward-trajectory' semi-Lagrangian
scheme. For multi-dimensional simulations of neutral gas flows, the
cell-centered version of this semi-Lagrangian (CCSL) scheme has advantages over
other options due to its implementation simplicity, low memory requirements,
and easier treatment of boundary conditions. The main drawback of the CCSL-CS
to date has been its high numerical diffusion in physical space, because of the
2$^{\text{nd}}$ order remapping that takes place at the end of each time step.
By means of a Modified Equation Analysis, it is shown that a high order
estimate of the remapping error can be obtained a priori, and a small
correction to the final position of the cells can be applied upon remapping, in
order to achieve full compensation of this error. The resulting scheme is
4$^{\text{th}}$ order accurate in space while retaining the desirable
properties of the CS: it is conservative and positivity-preserving, and the
overall algorithm complexity is not appreciably increased. Two monotone (i.e.
non-oscillating) versions of the fourth order CCSL-CS are also presented: one
uses a common flux-limiter approach; the other uses a non-polynomial
reconstruction to evaluate the derivatives of the density function. The method
is illustrated in simple one- and two-dimensional examples, and a fully 3D
solution of the Boltzmann equation describing expansion of a gas into vacuum
through a cylindrical tube.
