Two-dimensional Riemann problems for the compressible Euler system

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics; that is, simplified model is built via asymptotic analysis, numerical simulation and theoretical analysis. A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process. State-of-the-art numerical methods in numerical simulations are used to discern smallscale structures of the solutions, e. g., semi-hyperbolic patches. Analytical methods are used to establish the validity of the structure revealed in the numerical simulation. The entire process, used in many of Majda's programs, is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws.

Original languageEnglish (US)
Pages (from-to)845-858
Number of pages14
JournalChinese Annals of Mathematics. Series B
Volume30
Issue number6
DOIs
StatePublished - Dec 1 2009

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Euler System
Cauchy Problem
Computer simulation
Asymptotic analysis
Numerical Simulation
Pressure gradient
Conservation
Numerical methods
Gradient System
Systems of Conservation Laws
Pressure Gradient
Simulation Analysis
Applied mathematics
Analytical Methods
Asymptotic Analysis
Patch
Numerical Analysis
Theoretical Analysis
Numerical Methods
Entire

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics; that is, simplified model is built via asymptotic analysis, numerical simulation and theoretical analysis. A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process. State-of-the-art numerical methods in numerical simulations are used to discern smallscale structures of the solutions, e. g., semi-hyperbolic patches. Analytical methods are used to establish the validity of the structure revealed in the numerical simulation. The entire process, used in many of Majda's programs, is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws.",
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Two-dimensional Riemann problems for the compressible Euler system. / Zheng, Yuxi.

In: Chinese Annals of Mathematics. Series B, Vol. 30, No. 6, 01.12.2009, p. 845-858.

Research output: Contribution to journalArticle

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