  We extend to multi-dimensions the work of [1], where new fully explicit
kinetic methods were built for the approximation of linear and non-linear
convection-diffusion problems. The fundamental principles from the earlier work
are retained: (1) rather than aiming for the desired equations in the strict
limit of a vanishing relaxation parameter, as is commonly done in the diffusion
limit of kinetic methods, diffusion terms are sought as a first-order
correction of this limit in a Chapman-Enskog expansion, (2) introducing a
coupling between the conserved variables within the relaxation process by a
specifically designed collision matrix makes it possible to systematically
match a desired diffusion. Extending this strategy to multi-dimensions cannot,
however, be achieved through simple directional splitting, as diffusion is
likely to couple space directions with each other, such as with shear viscosity
in the Navier-Stokes equations. In this work, we show how rewriting the
collision matrix in terms of moments can address this issue, regardless of the
number of kinetic waves, while ensuring conservation systematically. This
rewriting allows for introducing a new class of kinetic models called
\emph{regularized} models, simplifying the numerical methods and establishing
connections with Jin-Xin models. Subsequently, new explicit arbitrary
high-order kinetic schemes are formulated and validated on standard
two-dimensional cases from the literature. Excellent results are obtained in
the simulation of a shock-boundary layer interaction, validating their ability
to approximate the Navier-Stokes equations with kinetic speeds obeying nothing
but a subcharacteristic condition along with a hyperbolic constraint on the
time step.
