Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling
The authors propose a new version of the Godunov scheme for the numerical approximation of solutions of a continuous traffic flow model. The phenomenon of phase transitions between a free flow (described by a scalar conservation law) and a congested flow (described by a \(2\times 2\) system) is taken into account. From an analytical point of view the mathematical model is well posed for all initial data with bounded total variation. The proposed numerical model is based on a modified averaging strategy and a sampling procedure. The mesh cells are modified following the phase boundaries and as such the projection involves only values belonging to the same phase. In order to come back to the original cells, the projection step is completed with a Glimm-type sampling technique. The accuracy of the scheme is extended to second order, in space and time, which is \(L^1\)-stable in time, using a MUSCL technique and a second order Runge-Kutta method. The validity and convergence of the method is demonstrated by several numerical tests.
scientific article; zbMATH DE number 5311226