# Preprint 2007-018

# Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model

## Raimund Bürger, Antonio García, Kenneth H. Karlsen and John D. Towers

**Abstract:**
The classical Lighthill–Witham–Richards (LWR)
kinematic traffic model is extended to a
unidirectional road on which the maximum density
*a*(*x*) represents road inhomogeneities,
such as variable numbers of lanes,
and is allowed to vary discontinuously.
The evolution of the car density
φ=φ(*x*,*t*)
can then be described by the initial value problem

(*) φ_{t}+(φ*v*(φ/*a*(*x*))_{x}=0, φ(*x*,0)=φ_{0}(*x*), *x*∈**R**, *t*∈(0,*T*).

Here *v*(*z*) is the velocity function,
where it is assumed that
*v*(*z*)≥0
and *v*(*z*) is nonincreasing.
Since *a*(*x*) is allowed to have a jump discontinuity,
(*) is a scalar conservation law with a spatially
discontinuous flux.
Herein we adapt to (*) the notion of entropy solutions of
type (*A*,*B*) put
forward in Bürger, Karlsen, Towers [Submitted, 2007],
which involves a Kružkov-type entropy inequality
based on a specific flux connection
(*A*,*B*).
We interpret our entropy theory in terms of traffic flow.
The driver's ride impulse, which has been used to justify
the standard Lax–Oleinik–Kružkov entropy
solutions when *a*(*x*) is constant,
cannot be used directly to determine the correct jump
condition at the interface where
*a*(*x*) is discontinuous.
We show by a parameter smoothing argument that our entropy
conditions are consistent with the driver's ride impulse.
Alternatively, we show that our notion of entropy solution
is consistent with the desire of drivers to speed up.
We prove that entropy solutions of type
(*A*,*B*) are
uniquely determined by their initial data.
Although other (equivalent) solution concepts exist,
the one used herein makes it possible to provide simple
and transparent convergence proofs for numerical schemes.
Indeed, we adjust to (*) a variant of the
Engquist–Osher (EO) scheme introduced recently in
Bürger, Karlsen, Towers [Submitted, 2007],
as well as a variant of the Hilliges–Weidlich (HW)
scheme analyzed by the authors in
Bürger, García, Karlsen, Towers [*J. Engrg. Math.*, to appear].
We improve the design, analysis, and performance of the HW scheme,
while maintaining its simplicity.
It is proven that these EO and HW schemes,
as well as a related Godunov scheme,
converge to the unique entropy solution of type
(*A*,*B*) of (*).
Via our entropy and compactness theory,
we give a unifying analysis of the three difference schemes.
In the case of the popular Godunov version of the scheme,
this represents the first convergence and well-posedness
result that is rigorous in that no unnecessarily
restrictive regularity assumptions are imposed on the
solution.
Results of numerical experiments are presented for first
order schemes and for MUSCL/Runge–Kutta versions
that are formally second order accurate.