We present a formulation of smoothed particle hydrodynamics (SPH) that
utilizes a first-order consistent reproducing kernel, a smoothing function that
exactly interpolates linear fields with particle tracers. Previous formulations
using reproducing kernel (RK) interpolation have had difficulties maintaining
conservation of momentum due to the fact the RK kernels are not, in general,
spatially symmetric. Here, we utilize a reformulation of the fluid equations
such that mass, linear momentum, and energy are all rigorously conserved
without any assumption about kernel symmetries, while additionally maintaining
approximate angular momentum conservation. Our approach starts from a
rigorously consistent interpolation theory, where we derive the evolution
equations to enforce the appropriate conservation properties, at the sacrifice
of full consistency in the momentum equation. Additionally, by exploiting the
increased accuracy of the RK method's gradient, we formulate a simple limiter
for the artificial viscosity that reduces the excess diffusion normally
incurred by the ordinary SPH artificial viscosity. Collectively, we call our
suite of modifications to the traditional SPH scheme Conservative Reproducing
Kernel SPH, or CRKSPH. CRKSPH retains many benefits of traditional SPH methods
(such as preserving Galilean invariance and manifest conservation of mass,
momentum, and energy) while improving on many of the shortcomings of SPH,
particularly the overly aggressive artificial viscosity and zeroth-order
inaccuracy. We compare CRKSPH to two different modern SPH formulations
(pressure based SPH and compatibly differenced SPH), demonstrating the
advantages of our new formulation when modeling fluid mixing, strong shock, and
adiabatic phenomena
