Numerical PDE Techniques for Scientists and Engineers

by: Prof. Dinshaw S. Balsara

Chapter 4: Non-Linear Conservation Laws; the Scalar Case

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Abstract

In this chapter we begin a study of non-linear conservation laws. To keep things simple, we focus on non-linear scalar conservation laws in this chapter. The next chapter will introduce us to the system case. After a gentle introduction to shock waves and rarefaction fans, we understand how these structures are a fundamental consequence of the non-linearity in the system. An intuitive study of shock waves precedes a study that shows them to be weak form solutions of the conservation law. Rarefaction fans are shown to be self-similar solutions of the nonlinear conservation law. We then design an approximate Riemann solver and show how it is applied to scalar conservation laws. The HLL Riemann solver described in this chapter turns out to be one of the standard building blocks for the numerical treatment of hyperbolic conservation laws.

Sections of the Chapter

• 4.1) Introduction
• 4.2) A Gentle Introduction to Rarefaction Waves and Shocks
  • 4.2.1) A Mechanistic Model for Rarefaction Waves and Shocks
  • 4.2.2) The Formation of Shocks and Rarefaction Waves
  • 4.2.3) Shock and Rarefaction Wave Solutions Arising from the Burgers Equation
  • 4.2.4) Simple Wave Solutions of the Burgers Equation
• 4.3) Isolated Shock Waves
  • 4.3.1) Shocks as Solutions of Viscous Equations that are Taken to the Inviscid Limit
  • 4.3.2) Shocks as Weak Solutions of a Hyperbolic Equation
• 4.4) Isolated Rarefaction Fans
  • 4.4.1) The Structure of an Isolated Rarefaction Fan
  • 4.4.2) The Role of Entropy in Arbitrating the Evolution of Discontinuities
• 4.5) The Entropy Fix and Approximate Riemann Solvers
  • 4.5.1) The Entropy Fix
  • 4.5.2) Approximate Riemann Solvers — The HLL and LLF Riemann Solvers
• 4.6) Boundary Conditions
• 4.7) Numerical Methods

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