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Macroscopic traffic flow continuum models are composed of
single or systems of partial difference equations (PDEs) with
suitable initial and boundary conditions which describe various
traffic phenomena and road geometry. These models have provided
a useful tool with which to test and assess road and traffic
control plans. Since the analytical solutions of traffic flow
continuum models are difficult to be solved. How to find an
approximate and efficient numerical solution becomes an
important course. This study takes aim at the numerical
finite difference methods of traffic flow continuum models. At
first, there are seven methods of explicit finite difference
schemes and two methods of implicit schemes used to solve the
first order linear continuum models, and compare the errors
between exact solution and numerical solutions of these methods.
In these results, there are three better algorithms, including
Lax-F、Lax-W and Leapfrog schemes, used to simulate the
quasilinear continuum models, and the Lax-F scheme can get a
more reasonable solution. In the process of numerical
computation, we found that every explicit finite difference
methods must satisfy the CFL condition to ensure the stability
and convergence. This condition requires the ratio of the mesh
in space and the mesh in time must satisfy some constrains, and
this makes the computation lack of efficiency. Therefore, this
study based on the CFL condition develops an adaptive finite
difference scheme to solve the LWR model more efficiently. This
adaptive scheme can determine automatically the nest suitable
time mesh size from the characteristic curve of every grid
points in this time, and it can converge to a stable solution.
In the numerical test, the Lax method and adaptive Lax method
are used to solve the LWR model with different initial and
boundary conditions. The simulation results show us that the
adaptive Lax method is more efficiency than the Lax method.
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