  We present a second-order formulation of the LWR model based on Phillips'
model (Phillips, 1979); but the model is nonstandard with a hyperreal
infinitesimal relaxation time. Since the original Phillips model is unstable
with three different definitions of stability in both Eulerian and Lagrangian
coordinates, we cannot use traditional methods to prove the equivalence between
the second-order model, which can be considered the zero-relaxation limit of
Phillips' model, and the LWR model, which is the equilibrium counterpart of
Phillips' model. Instead, we resort to a nonstandard method based on the
equivalence relationship between second-order continuum and car-following
models established in (Jin, 2016) and prove that the nonstandard model and the
LWR model are equivalent, since they have the same anisotropic car-following
model and stability property. We further derive conditions for the nonstandard
model to be forward-traveling and collision-free, prove that the collision-free
condition is consistent with but more general than the CFL condition (Courant
et al., 1928), and demonstrate that only anisotropic and symplectic Euler
discretization methods lead to physically meaningful solutions. We numerically
solve the lead-vehicle problem and show that the nonstandard second-order model
has the same shock and rarefaction wave solutions as the LWR model for both
Greenshields and triangular fundamental diagrams; for a non-concave fundamental
diagram we show that the collision-free condition, but not the CFL condition,
yields physically meaningful results. Finally we present a correction method to
eliminate negative speeds and collisions in general second-order models, and
verify the method with a numerical example.
