TY - JOUR

T1 - Two-dimensional Riemann problems for the compressible Euler system

AU - Zheng, Yuxi

N1 - Funding Information:
Manuscript received April 4, 2009. Published online August 19, 2009. ∗Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. E-mail: yzheng@math.psu.edu ∗∗Project supported partially by the National Science Foundation (No. DMS-0603859).

PY - 2009/12

Y1 - 2009/12

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=71349085576&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71349085576&partnerID=8YFLogxK

U2 - 10.1007/s11401-009-0114-5

DO - 10.1007/s11401-009-0114-5

M3 - Article

AN - SCOPUS:71349085576

SN - 0252-9599

VL - 30

SP - 845

EP - 858

JO - Chinese Annals of Mathematics. Series B

JF - Chinese Annals of Mathematics. Series B

IS - 6

ER -