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Energy versus entropy estimates for nonlinear hyperbolic systems of equations
Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.ORCID-id: 0000-0002-7972-6183
Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.ORCID-id: 0000-0002-5902-1522
2019 (engelsk)Rapport (Annet vitenskapelig)
Abstract [en]

We compare and contrast information provided by the energy analysis of Kreiss and the entropy theory of Tadmor for systems of nonlinear hyperbolic conservation laws. The two-dimensional nonlinear shallow water equations are used to highlight the similarities and differences since the total energy of the system is a mathematical entropy function. We demonstrate that the classical energy method is consistent with the entropy analysis, but significantly more fundamental as it guides proper boundary treatments. In particular, the energy analysis provides information on what type of and how many boundary conditions are required, which is lacking in the entropy analysis. For the shallow water system we determine the number and the type of boundary conditions needed for subcritical and supercritical flows on a general domain. As eigenvalues are augmented in the nonlinear analysis, we find that a flow may be classified as subcritical, but the treatment of the boundary resembles that of a supercritical flow. Because of this, we show that the nonlinear energy analysis leads to a different number of boundary conditions compared with the linear energy analysis. We also demonstrate that the entropy estimate leads to erroneous boundary treatments by over specifying and/or under specifying boundary data causing the loss of existence and/or energy bound, respectively. Our analysis reveals that the nonlinear energy analysis is the only one that provides an estimate for open boundaries. Both the entropy and linear energy analysis fail.

sted, utgiver, år, opplag, sider
Linköping: Linköping University Electronic Press, 2019. , s. 23
Serie
LiTH-MAT-R, ISSN 0348-2960 ; 2019:8
HSV kategori
Identifikatorer
URN: urn:nbn:se:liu:diva-159114ISRN: LiTH-MAT-R--2019/08--SEOAI: oai:DiVA.org:liu-159114DiVA, id: diva2:1339295
Tilgjengelig fra: 2019-07-29 Laget: 2019-07-29 Sist oppdatert: 2019-08-13

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Energy versus entropy estimates for nonlinear hyperbolic systems of equations(591 kB)18 nedlastinger
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