A Shock-Fitting Primer

A defining feature of nonlinear hyperbolic equations is the occurrence of shock waves. While the popular shock-capturing methods are easy to implement, shock-fitting techniques provide the most accurate results. A Shock-Fitting Primer presents the proper numerical treatment of shock waves and other discontinuities.

The book begins by recounting the events that lead to our understanding of the theory of shock waves and the early developments related to their computation. After presenting the main shock-fitting ideas in the context of a simple scalar equation, the author applies Colombeau’s theory of generalized functions to the Euler equations to demonstrate how the theory recovers well-known results and to provide an in-depth understanding of the nature of jump conditions. He then extends the shock-fitting concepts previously discussed to the one-dimensional and quasi-one-dimensional Euler equations as well as two-dimensional flows. The final chapter explores existing and future developments in shock-fitting methods within the framework of unstructured grid methods.

Throughout the text, the techniques developed are illustrated with numerous examples of varying complexity. On the accompanying CD-ROM, MATLAB® codes serve as concrete examples of how to implement the ideas discussed in the book.

Manuel D. Salas is a distinguished research associate at NASA Langley Research Center in Hampton, Virginia, USA. During his tenure at NASA, Mr. Salas was head of the theoretical aerodynamics branch, chief scientist for fluid dynamics, director of high performance computing, and principal investigator for the hypersonic aerodynamic program. He was also director of the Institute for Computer Applications in Science and Engineering (ICASE) from 1996 to 2002.

Introduction
Prelude
The Curious Events Leading to the Theory of Shock Waves
Early Attempts at Computing Flows with Shocks
Shock-Fitting Principles
The Inviscid Burgers’ Equation
The One-Saw-Tooth Problem
Background Numerical Schemes
Mappings, Conservation Form, and Transformation Matrices
Boundary Shock-Fitting
Gaussian Pulse Problem
Boundary Shock-Fitting Revisited
Floating Shock-Fitting
Detection of Shock Formation
Application of Colombeau’s Generalized Functions to a Nonconservative System of Equations
Fundamental Concepts and Equations
Physical Problem
Mathematical Formulation
Explicit Form of the Equations of Motion
Orthogonal Curvilinear Coordinates
Differential Geometry of Singular Surfaces
Finite Discontinuities
Shock Wave Structure
Euler Equations: One-Dimensional Problems
Piston-Driven Flows
Numerical Analysis of a Simple Wave Region
Shock Wave Computation
Quasi-One-Dimensional Flows
Euler Equations: Two-Dimensional Problems
The Blunt Body Problem
External Conical Corners
Supersonic Flow over Elliptical Wings
Floating Shock-Fitting with Unstructured Grids
Introduction
Unstructured Grids: Preliminaries
Unstructured Grid Solver
Application to Euler Equations
Floating Shock-Fitting Implementation
Unstructured Grids Shock-Fitting Results
References
Appendix
Index
Problems appear at the end of each chapter.