  We introduce a deterministic discrete-particle simulation approach, the
Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear
deformations of the particles to reduce the noise traditionally associated with
particle schemes. Formally, transforming the particles is justified by local
first order expansions of the characteristic flow in phase space. In practice
the method amounts to using deformation matrices within the particle shape
functions; these matrices are updated via local evaluations of the forward
numerical flow. Because it is necessary to periodically remap the particles on
a regular grid to avoid excessively deforming their shapes, the method can be
seen as a development of Denavit's Forward Semi-Lagrangian (FSL) scheme [J.
Denavit, J. Comp. Physics 9, 75 (1972)]. However, it has recently been
established [M. Campos Pinto, "Smooth particle methods without smoothing",
arXiv:1112.1859 (2012)] that the underlying Linearly-Transformed Particle
scheme converges for abstract transport problems, with no need to remap the
particles; deforming the particles can thus be seen as a way to significantly
lower the remapping frequency needed in the FSL schemes, and hence the
associated numerical diffusion. To couple the method with electrostatic field
solvers, two specific charge deposition schemes are examined, and their
performance compared with that of the standard deposition method. Finally,
numerical 1d1v simulations involving benchmark test cases and halo formation in
an initially mismatched thermal sheet beam demonstrate some advantages of our
LTPIC scheme over the classical PIC and FSL methods. Benchmarked test cases
also indicate that, for numerical choices involving similar computational
effort, the LTPIC method is capable of accuracy comparable to or exceeding that
of state-of-the-art, high-resolution Vlasov schemes.
