# Numerical solution of the 1D viscous Burgers' and traffic flow equations by the inflow-implicit/outflow-explicit finite volume method

## Main Article Content

## Abstract

In this article we solve numerically the one-dimensional viscous Burgers' equation by the inflow-implicit/outflow-explicit method. The method is based on finite volume space discretization and a semi-implicit discretization in time. Inflows to the cells are treated implicitly and outflows explicitly. Comparisons of numerical solutions with the exact ones are presented. As a physical interpretation of the Burgers' equation we chose a simple continuum traffic flow model.

## Article Details

How to Cite

Ibolya, G., & Mikula, K.
(2020).
Numerical solution of the 1D viscous Burgers' and traffic flow equations by the inflow-implicit/outflow-explicit finite volume method.

*Proceedings Of The Conference Algoritmy,*, 191 - 200. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1583/836
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Articles

## References

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[2] P. J. Olver, Introduction to Partial Differential Equations, Undergraduate Texts in Mathematics. Springer, New York, 2014.

[3] K. Mikula, M. Ohlberger, A new Inflow-Implicit/Outflow-Explicit Finite Volume Method for Solving Variable Velocity Advection Equations, Preprint 01/10 - N, Angewandte Mathematik und Informatik, Universitaet Münster, June 2010

[4] K. Mikula, M. Ohlberger, Inflow-Implicit/Outflow-Explicit Scheme for Solving Advection Equations, in Finite Volumes in Complex Applications VI, Problems & Perspectives, Eds.J.For̆t et al. (Proceedings of the Sixth International Conference on Finite Volumes in Complex Applications, Prague, June 6-10, 2011), Springer Verlag, 2011, pp. 683-692.

[5] K. Mikula, M. Ohlberger, J.Urban, Inflow-Implicit/Outflow-Explicit finite volume methods for solving advection equations, Applied Numerical Mathematics, Vol. 85 (2014) pp. 16-37

[6] B. N. Greenshields, A study of traffic capacity, In Proceedings of the 14th Annual Meeting of the Highway Research Board, 1934, pp. 448-474.

[7] M. J. Lighthill and G. B. Whitham, On kinematic waves II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, (1955), pp. 317-345.

[8] P. I. Richards, Shock waves on highways, Oper. Res., 4 (1956), pp. 42-51.

[9] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974. 2

[2] P. J. Olver, Introduction to Partial Differential Equations, Undergraduate Texts in Mathematics. Springer, New York, 2014.

[3] K. Mikula, M. Ohlberger, A new Inflow-Implicit/Outflow-Explicit Finite Volume Method for Solving Variable Velocity Advection Equations, Preprint 01/10 - N, Angewandte Mathematik und Informatik, Universitaet Münster, June 2010

[4] K. Mikula, M. Ohlberger, Inflow-Implicit/Outflow-Explicit Scheme for Solving Advection Equations, in Finite Volumes in Complex Applications VI, Problems & Perspectives, Eds.J.For̆t et al. (Proceedings of the Sixth International Conference on Finite Volumes in Complex Applications, Prague, June 6-10, 2011), Springer Verlag, 2011, pp. 683-692.

[5] K. Mikula, M. Ohlberger, J.Urban, Inflow-Implicit/Outflow-Explicit finite volume methods for solving advection equations, Applied Numerical Mathematics, Vol. 85 (2014) pp. 16-37

[6] B. N. Greenshields, A study of traffic capacity, In Proceedings of the 14th Annual Meeting of the Highway Research Board, 1934, pp. 448-474.

[7] M. J. Lighthill and G. B. Whitham, On kinematic waves II: A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A, (1955), pp. 317-345.

[8] P. I. Richards, Shock waves on highways, Oper. Res., 4 (1956), pp. 42-51.

[9] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974. 2