Numerical PDE Techniques for Scientists and Engineers

by: Prof. Dinshaw S. Balsara

Chapter 3: Scalar Advection and Linear Hyperbolic Systems

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Abstract

The previous chapter showed us how we can construct a finite difference approximation (FDA) for any partial differential equation (PDE). Our study of consistency and stability showed us that we can construct FDAs for PDEs and be assured that for well-specified initial and boundary conditions they will converge to the actual solution of the PDE. However, when we applied our ideas to the scalar advection equation we discovered several deficiencies in every linear FDA that we constructed for solving that equation. In this chapter, we understand scalar advection in two easy stages. We first show that there is a pictorial approach to advection that provides several important insights. The insights then enable us to overcome the previously-mentioned deficiencies. We then formalize those insights. We show that it is possible to define some general properties for a second order accurate scheme for numerical advection which ensure that the solution will not generate undesirable oscillations.

We then move on to examine linear hyperbolic systems of equations. Viewed in the space of characteristic variables, we see that a study of linear hyperbolic systems can be turned into a study of a set of scalar advection equations. The simple wave solutions that are supported by linear hyperbolic systems are close analogues of shocks which we will study in the next chapter. This chapter also gives us our first introduction to the Riemann problem. We see that it is an essential building block that is routinely used in numerical methods for hyperbolic conservation laws.

Boundary conditions for hyperbolic systems are discussed. The chapter ends with a description of several popular numerical schemes for linear hyperbolic systems. We focus on the a modified Lax-Wendroff scheme, the two-step Runge-Kutta scheme and the predictor-corrector scheme.

Sections of the Chapter

• 3.1) Introduction
• 3.2) Qualitative Introduction to Non-Linear Hybridization for Scalar Advection (Slides 1, Slides 2, Slides 3)
• 3.3) The Total Variation Diminishing Property and Understanding the Limiters (Slides 1, Slides 2)
• 3.4) Linear Hyperbolic Systems and the Riemann Problem
  • 3.4.1) Solution of Linear Hyperbolic PDEs for Continuous Initial Data
  • 3.4.2) Solution of Linear Hyperbolic PDEs for Discontinuous Initial Data: Simple Waves and the Riemann Problem
  • 3.4.3) The Riemann Problem as a Building Block for the Numerical Solution of Hyperbolic Systems
• 3.5) Numerical Boundary Conditions for Linear Hyperbolic Systems
• 3.6) Second Order Upwind Schemes for Linear Hyperbolic Systems
  • 3.6.1) Second Order Accurate Extension of the Lax-Wendroff Scheme with Limiters
  • 3.6.2) Second Order Accurate, Two-Stage Runge-Kutta Scheme with Limiters
  • 3.6.3) Predictor-Corrector Formulation
  • 3.6.4) Numerical Results from the Previous Three Schemes

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