On the choice of shock capturing schemes for the solution of the LWR traffic flow equation using a high order modal discontinuous Galerkin discretization

Abstract

In this paper we studied different techniques to stabilize the numerical solution of the LWR traffic flow equation when using a modal version of the high order discontinuous Galerkin finite element method (DGFEM). Based on a set of three standard examples of traffic flow initial conditions, we compared four shock capturing strategies to control Gibbs phenomenon around discontinuous solutions of the car density conserved variable: (1) generalized slope limiter (ΠN), (2) hierarchical slope limiter, (3) sub-cell shock capturing with elementwise constant artificial viscosity, and (4) sub-cell shock capturing with local C0 artificial viscosity. Although such stabilizing techniques were well documented for aerospace applications, our experiments revealed important features which are worth mentioning. Firstly, regardless of the polynomial order of the approximation, both limiters required quite refined meshes to achieve sharp shock resolution. Secondly, shock capturing based on elementwise constant viscosity produced small traveling spikes over the solution profiles destroying the locality of the method and resulting in excessive smearing of the discontinuities. Finally, C0 artificial viscosity addition produced sharp shock resolutions with very coarse meshes and high order polynomial approximations. Overall, this technique demonstrated the better cost-benefit in terms of computational effort (number of degrees of freedom) to achieve a certain level of shock resolution.