  We present a new class of particle methods with deformable shapes that
converge in the uniform norm without requiring remappings, extended overlapping
or vanishing moments for the particles. The crux of the method is to use
polynomial expansions of the backward characteristic flow to transport the
numerical particles with improved accuracy: in the first order case the method
consists of representing the transported density with linearly-transformed
particles, the second order version computes quadratically-transformed
particles, and so on. For programming purposes we provide explicit
implementations of the resulting LTP and QTP schemes that only involve
pointwise evaluations of the forward characteristic flow, and also come with
local indicators for the accuracy of the corresponding transport scheme.
Numerical tests using different transport problems demonstrate the accuracy of
the proposed methods compared to standard particle schemes, and establish their
robustness with respect to the remapping period. In particular, it is shown
that QTP particles can be transported without remappings on very long periods
of times, without hampering the accuracy of the numerical solutions. Finally, a
dynamic criterion is proposed to automatically select the time steps where the
particles should be remapped. The strategy does not require additional
inter-particle communications, and it is validated by numerical experiments.
