Two-Dimensional Riemann Problems for the Compressible Euler System |
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Authors: | Yuxi ZHENG |
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Institution: | Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA |
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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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Keywords: | Characteristic decomposition Guderley reflection Hodograph transform Pressure gradient system Self-similar Semi-hyperbolic wave Triple point paradox Riemann problem Riemann variable |
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