The pressure gradient system
type: Item
instance of: scholarly article
title: The pressure gradient system
zbMATH Open document ID: 1227.35208
author: Yuxi Zheng
author: Zachary Robinson
published in: Methods and Applications of Analysis
publication date: 2011-07-22T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.maa/1306182406
review text: The authors of this very interesting paper consider the two-dimensional compressible Euler system \[ \begin{aligned} \rho_t&+\nabla\cdot (\rho {\mathbf u})=0,\\ (\rho{\mathbf u})_t&+\nabla\cdot (\rho{\mathbf u}\times{\mathbf u}+pI)=0,\\ (\rho E)_t&+\nabla\cdot (\rho E{\mathbf u}+ p{\mathbf u})=0, \end{aligned} \] where \(\rho \) is the density, \({\mathbf u}\) is the velocity vector, \(p\) is the pressure, \(E=|{\mathbf u}|^2/2+p\rho^{-1}/(\gamma -1)\) is the total energy density per unit mass, and \(\gamma >1\) is the gas constant. The so-called pressure gradient system is a subsystem of the compressible Euler system \[ u_t+p_x=0, \quad v_t+p_y=0, \quad E_t+(up)_x+(vp)_y=0, \] where \(E=p+(u^2+v^2)/2\). The authors provide numerical simulations, basic characteristic analysis and physical considerations for the Riemann problems to the considered model to find out appropriate internal conditions at the origin. This study reveals subtle structures of the velocity. Both components exhibit discontinuities at the origin. Comparing to the roll-up of shear waves or vortex-sheets of the Euler system, it turns out that the singularities are mild and occur only along rays from the origin. Numerical examples illustrate the theory, and sets of central contour lines from the pressure graph are shown in a very precise form.
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35Q31
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 76N15
zbMATH DE Number: 5930178
zbMATH Keywords: dimension splitting
zbMATH Keywords: Riemann problem
zbMATH Keywords: Roe approximation
zbMATH Keywords: compressible Euler system
zbMATH Keywords: two space dimensions
zbMATH Keywords: roll-up of waves
MaRDI profile type: Publication
OpenAlex ID: W2081186122
DOI: 10.4310/MAA.2010.V17.N3.A2
description: scientific article; zbMATH DE number 5930178