zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An unsplit, higher order Godunov method for scalar conservation laws in multiple dimensions. (English) Zbl 0684.65088

The authors develop an unsplit higher order Godunov method for scalar conservation laws in two dimensions. The method represents an extension of methods previously developed by P. Collela [A multidimensional second order Godunov scheme for conservation laws (to appear)] and B. van Leer [Computing methods in applied sciences and engineering VI, Proc. 6th Int. Symp., Versailles 1983, 493-497 (1984; Zbl 0565.65052)]. The resulting method is shown to satisfy a maximum principle for constant coefficient linear advection. Tests of the method on a variety of linear advection problems indicate that the method is more accurate than existing methods of this type. Although the improvement for the propagation of a pure discontinuity is rather modest, the improvement for smooth structure is more substantial.

In particular, the method does a better job of preserving shape of the profile as it is propagated than other methods. The major difficulty with the scheme is its complexity. This renders the method costly for general application. For applications to e.g. porous media flow the computational cost is dominated by the solution of the elliptic pressure equation. For this type of equation where a conservation law is solved as a part of a larger computational tast, the complexity of the scheme does not present a problem.

Reviewer: Ph.Brenner

MSC:
65M06Finite difference methods (IVP of PDE)
76S05Flows in porous media; filtration; seepage
35L65Conservation laws