High-Order Numerical Schemes for One-Dimensional Nonlocal Conservation Laws

Abstract

This paper focuses on the numerical approximation of the solutions of non-local conservation laws in one space dimension. These equations are motivated by two distinct applications, namely a traffic flow model in which the mean velocity depends on a weighted mean of the downstream traffic density, and a sedimentation model where either the solid phase velocity or the solid-fluid relative velocity depends on the concentration in a neighborhood. In both models, the velocity is a function of a convolution product between the unknown and a kernel function with compact support. It turns out that the solutions of such equations may exhibit oscillations that are very difficult to approximate using classical first-order numerical schemes. We propose to design Discontinuous Galerkin (DG) schemes and Finite Volume WENO (FV-WENO) schemes to obtain high-order approximations. As we will see, the DG schemes give the best numerical results but their CFL condition is very restrictive. On the contrary, FV-WENO schemes can be used with larger time steps. We will see that the evaluation of the convolution terms necessitates the use of quadratic polynomials reconstructions in each cell in order to obtain the high-order accuracy with the FV-WENO approach. Simulations using DG and FV-WENO schemes are presented for both applications.

A Godunov type scheme for a class of LWR traffic flow models with non-local flux

“…The latter will be briefly introduced in the following section. In addition, we also comment on the differences between the model considered in this work (4) to (6) and the earlier one (1) to (3).…”

“…Our main result concerning the new model is given by the following theorem, which states the well-posedness of problem (4) to (6). Theorem 2.3.…”

“…Theorem 2.4. Under hypotheses (H2), let ρ and σ be two entropy solutions of (4) to (6) with initial data ρ 0 and σ 0 , respectively. Then, for any T > 0, there holds…”

Maximum Principle Satisfying CWENO Schemes for Nonlocal Conservation Laws

“…To evaluate an approximation of the right-hand side of (5), we will make use of CWENO reconstructions based on the cell averagesρ j (t), which is described in the following section. Afterwards, we explain all further ingredients to achieve a fully discrete scheme, beginning with the choice of a suitable quadrature rule to approximate the initial cell averages (7) and to evaluate the convolution term contained in (5).…”

“…To achieve the desired orders of accuracy, we apply the Radau-Legendre quadrature rules with R = 2, 3, 4 for CWENO3 (g = 1), CWENO5 (g = 2) and CWENO7 (g = 3), respectively. Note that without regard to the maximum principle, which comes with a more restrictive CFL condition, one could as well apply any other quadrature rule of sufficient accuracy to approximate the initial cell averages and to evaluate the convolution term -like the Gauß-Legendre formulas as used in [7].…”