Numerical PDE Techniques for Scientists and Engineers

by: Prof. Dinshaw S. Balsara

Chapter 5: The Hydrodynamical Riemann Problem

• Abstract
• Movies
• Slides with Voice-Over
• Chapters, Codes, Etc
• Textbook Home

Abstract

Having studied shocks and rarefaction fans for non-linear scalar conservation laws in the previous chapter, we now move on to a study of the same for systems of hyperbolic conservation laws. We use the Euler equations as an example. Discontinuous solutions of the Euler equations - shocks and contact discontinuities - are studied in detail. We then study isentropic flow and rarefaction fans in detail. The Riemann problem is shown to be an essential building block for the numerical solution of systems of hyperbolic conservation laws. The solution of the hydrodynamical Riemann problem is presented in detail. It is applied to several shock tube problems and the results are discussed.

Sections of the Chapter

• 5.1) Introduction
  • 5.1.1) General Introduction to the Riemann Problem
  • 5.1.2) Introducing The Riemann Problem as a Building Block for Numerical Schemes
• 5.2) Hydrodynamical Shock Waves
  • 5.2.1) Hydrodynamical Discontinuities and Shock Jump Conditions
  • 5.2.2) The Hugoniot Adiabat
  • 5.2.3) Normal Shocks in Polytropic Gases
  • 5.2.4) Shocks in the Lab Frame
• 5.3) Rarefaction Fans (Slides 1, Slides 2)
  • 5.3.1) Right-going Rarefaction Fans
  • 5.3.2) Left-going Rarefaction Fans
• 5.4) The Riemann Problem
  • 5.4.1) Intuitive Introduction to the Riemann Problem
  • 5.4.2) Iterative Solution of the Riemann Problem

© 2012 - All Rights Reserved