A random sampling method for a family of Templeclass systems of conservation laws
(2018) In Numerische Mathematik 138(1). p.3773 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 wellknown 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 wellknown 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 nonuniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.
(Less)
 author
 Betancourt, Fernando; Bürger, Raimund; Chalons, Christophe; Diehl, Stefan ^{LU} and Farås, Sebastian ^{LU}
 organization
 publishing date
 201801
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Numerische Mathematik
 volume
 138
 issue
 1
 pages
 37  73
 publisher
 Springer
 external identifiers

 scopus:85023775470
 ISSN
 0029599X
 DOI
 10.1007/s002110170900z
 language
 English
 LU publication?
 yes
 id
 c89e7060704b45f7848695efddfa60e2
 date added to LUP
 20170727 09:58:26
 date last changed
 20180219 02:08:44
@article{c89e7060704b45f7848695efddfa60e2, 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 wellknown 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 nonuniqueness 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 = {0029599X}, language = {eng}, number = {1}, pages = {3773}, publisher = {Springer}, series = {Numerische Mathematik}, title = {A random sampling method for a family of Templeclass systems of conservation laws}, url = {http://dx.doi.org/10.1007/s002110170900z}, volume = {138}, year = {2018}, }