  Recently different formulations of the first-order Lighthill-Whitham-Richards
(LWR) model have been identified in different coordinates and state variables.
However, there exists no systematic method to convert higher-order continuum
models into car-following models and vice versa. In this study we propose a
simple method to enable systematic conversions between higher-order continuum
and car-following models in two steps: equivalent transformations of variables
between Eulerian and Lagrangian coordinates, and finite difference
approximations of first-order derivatives in Lagrangian coordinates. With the
method, we derive higher-order continuum models from a number of well-known
car-following models. We also derive car-following models from higher-order
continuum models. For general second-order models, we demonstrate that the
car-following and continuum formulations share the same fundamental diagram,
but the string stability condition of a car-following model is different from
the linear stability condition of a continuum model. This study reveals
relationships between many existing models and also leads to a number of new
models.
