Based on an established mathematical model for the behavior of large crowds, a new model is derived that is able to take into account the statistical variation of individual maximum walking speeds. The same model is shown to be valid also in traffic flow situations, where for instance the statistical variation of preferred maximum speeds can be considered. The model involves explicit bounds on the state variables, such that a special Riemann solver is derived that is proved to respect the state constraints. Some care is devoted to a valid construction of random initial data, necessary for the use of the new model. The article also includes a numerical method that is shown to respect the bounds on the state variables and illustrative numerical examples, explaining the properties of the new model in comparison with established models.
Citation: Veronika Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties[J]. Mathematical Biosciences and Engineering, 2015, 12(2): 393-413. doi: 10.3934/mbe.2015.12.393
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Abstract
Based on an established mathematical model for the behavior of large crowds, a new model is derived that is able to take into account the statistical variation of individual maximum walking speeds. The same model is shown to be valid also in traffic flow situations, where for instance the statistical variation of preferred maximum speeds can be considered. The model involves explicit bounds on the state variables, such that a special Riemann solver is derived that is proved to respect the state constraints. Some care is devoted to a valid construction of random initial data, necessary for the use of the new model. The article also includes a numerical method that is shown to respect the bounds on the state variables and illustrative numerical examples, explaining the properties of the new model in comparison with established models.
References
|
[1]
|
preprint, 2013.
|
|
[2]
|
SIAM J. Appl. Math., 60 (2000), 916-938 (electronic).
|
|
[3]
|
Math. Models Methods Appl. Sci., 18 (2008), 1317-1345.
|
|
[4]
|
SIAM Review, 53 (2011), 409-463.
|
|
[5]
|
Mathematical Models and Methods in the Applied Sciences, 22 (2012), 1150023, 34 p.
|
|
[6]
|
SIAM J. Appl. Dyn. Syst., 11 (2012), 741-770.
|
|
[7]
|
Numer. Math., 34 (1980), 285-314.
|
|
[8]
|
Multiscale Model. Simul., 9 (2011), 155-182.
|
|
[9]
|
Transp. Res. B, 29 (1995), 277-286.
|
|
[10]
|
J. Math. Pures Appl. (9), 74 (1995), 483-548.
|
|
[11]
|
The European Physical Journal B, 69 (2009), 539-548.
|
|
[12]
|
Modelling and Optimisation of Flows on Networks, (2013), 271-302.
|
|
[13]
|
Transportation Research Part B: Methodological, 36 (2002), 507-535.
|
|
[14]
|
Mat. Sb. (N.S.), 81 (1970), 228-255.
|
|
[15]
|
Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.
|
|
[16]
|
Math. Comp., 81 (2012), 1979-2018.
|
|
[17]
|
Operations Res., 4 (1956), 42-51.
|
|
[18]
|
IEEE Transactions on Pattern Analysis and Machine Intelligence, 29 (2007), 1563-1574.
|
|
[19]
|
New Journal of Physics, 10 (2008), 033001.
|
|
[20]
|
New Journal of Physics, 15 (2013), 103034.
|
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