Jan Friedrich

Jan Friedrich, Oliver Kolb, Simone Göttlich

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide $L^\infty$ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

Jan Friedrich, Oliver Kolb

Central WENO schemes are a natural candidate for higher-order schemes for non-local conservation laws, since the underlying reconstructions do not only provide single point values of the solution but a complete (high-order) reconstruction in every time step, which is beneficial to evaluate the integral terms. Recently, in [C. Chalons et al., SIAM J. Sci. Comput., 40(1), A288-A305], Discontinuous Galerkin (DG) schemes and Finite Volume WENO (FV-WENO) schemes have been proposed to obtain high-order approximations for a certain class of non-local conservation laws. In contrast to their schemes, the presented CWENO approach neither requires a very restrictive CFL condition (as the DG methods) nor an additional reconstruction step (as the FV-WENO schemes). Further, by making use of the well-known linear scaling limiter of [X. Zhang and C.-W. Shu, J. Comput. Phys., 229, p. 3091-3120], our CWENO schemes satisfy a maximum principle for suitable non-local conservation laws.