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 $L^∞$ 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.
Citation: Jan Friedrich, Oliver Kolb, Simone Göttlich. A Godunov type scheme for a class of LWR traffic flow models with non-local flux[J]. Networks and Heterogeneous Media, 2018, 13(4): 531-547. doi: 10.3934/nhm.2018024
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 $L^∞$ 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.
| [1] |
Nonlocal systems of conservation laws in several space dimensions. SIAM J. Numer. Anal. (2015) 53: 963-983. 
|
| [2] |
On the numerical integration of scalar nonlocal conservation laws. ESAIM Math. Model. Numer. Anal. (2015) 49: 19-37.
|
| [3] |
A continuum model for a re-entrant factory. Oper. Res. (2006) 54: 933-950.
|
| [4] |
On nonlocal conservation laws modelling sedimentation. Nonlinearity (2011) 24: 855-885.
|
| [5] |
Well-posedness of a conservation law with non-local flux arising in traffic flow modeling. Numer. Math. (2016) 132: 217-241.
|
| [6] |
High-order numerical schemes for one-dimensional nonlocal conservation laws. SIAM J. Sci. Comput. (2018) 40: A288-A305.
|
| [7] |
Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. ESAIM: Mathematical Modelling and Numerical Analysis (2018) 52: 163-180.
|
| [8] |
A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. (2012) 22: 1150023-1150057.
|
| [9] |
Control of the continuity equation with a non local flow. ESAIM Control Optim. Calc. Var. (2011) 17: 353-379.
|
| [10] | P. Goatin and S. Scialanga, The Lighthill-Whitham-Richards traffic flow model with non-local velocity: Analytical study and numerical results, INRIA Research Report, 2015. |
| [11] |
Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Netw. Heterog. Media (2016) 11: 107-121.
|
| [12] |
Modeling, simulation and validation of material flow on conveyor belts. Appl. Math. Model. (2014) 38: 3295-3313.
|
| [13] |
S. N. Kružkov, First order quasilinear equations in several independent variables,
Mathematics of the USSR-Sbornik, 10 (1970), 217. doi: 10.1070/SM1970v010n02ABEH002156
|
| [14] |
R. J. LeVeque,
Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-5116-9
|
| [15] |
On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. (1955) 229: 317-345.
|
| [16] |
Shock waves on the highway. Oper. Res. (1956) 4: 42-51.
|
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Jan Friedrich, Oliver Kolb, Simone Göttlich. A Godunov type scheme for a class of LWR traffic flow models with non-local flux[J]. Networks and Heterogeneous Media, 2018, 13(4): 531-547. doi: 10.3934/nhm.2018024
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