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A random sampling method for a family of Temple-class systems of conservation laws

Betancourt, Fernando; Bürger, Raimund; Chalons, Christophe; Diehl, Stefan LU and Farås, Sebastian LU (2017) In Numerische Mathematik
Abstract

The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear (Formula presented.) systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between... (More)

The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear (Formula presented.) systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between averaging (Av) and remap steps similar to the approach by Chalons and Goatin (Commun Math Sci 5:533–551, 2007) generates numerical solutions that do satisfy an invariant region property. If the remap step is made by random sampling (RS), then combined techniques due to Glimm (Commun Pure Appl Math 18:697–715, 1965), LeVeque and Temple (Trans Am Math Soc 288:115–123, 1985) prove that the resulting Av–RS scheme converges to a weak solution. Numerical examples illustrate that the new scheme is superior to Godunov’s method in accuracy and resolution. Secondly, the vacuum state, which may form even from positive initial data, causes potential problems of non-uniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.

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publication status
epub
subject
in
Numerische Mathematik
pages
37 pages
publisher
Springer
external identifiers
  • scopus:85023775470
ISSN
0029-599X
DOI
10.1007/s00211-017-0900-z
language
English
LU publication?
yes
id
c89e7060-704b-45f7-8486-95efddfa60e2
date added to LUP
2017-07-27 09:58:26
date last changed
2017-07-27 09:58:26
@article{c89e7060-704b-45f7-8486-95efddfa60e2,
  abstract     = {<p>The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear (Formula presented.) systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between averaging (Av) and remap steps similar to the approach by Chalons and Goatin (Commun Math Sci 5:533–551, 2007) generates numerical solutions that do satisfy an invariant region property. If the remap step is made by random sampling (RS), then combined techniques due to Glimm (Commun Pure Appl Math 18:697–715, 1965), LeVeque and Temple (Trans Am Math Soc 288:115–123, 1985) prove that the resulting Av–RS scheme converges to a weak solution. Numerical examples illustrate that the new scheme is superior to Godunov’s method in accuracy and resolution. Secondly, the vacuum state, which may form even from positive initial data, causes potential problems of non-uniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.</p>},
  author       = {Betancourt, Fernando and Bürger, Raimund and Chalons, Christophe and Diehl, Stefan and Farås, Sebastian},
  issn         = {0029-599X},
  language     = {eng},
  month        = {07},
  pages        = {37},
  publisher    = {Springer},
  series       = {Numerische Mathematik},
  title        = {A random sampling method for a family of Temple-class systems of conservation laws},
  url          = {http://dx.doi.org/10.1007/s00211-017-0900-z},
  year         = {2017},
}