A Shock-Fitting Primer  book cover
1st Edition

A Shock-Fitting Primer

ISBN 9781138116634
Published June 14, 2017 by CRC Press
416 Pages 170 B/W Illustrations

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Book Description

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 downloadable resources, MATLAB® codes serve as concrete examples of how to implement the ideas discussed in the book.

Table of Contents



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


Unstructured Grids: Preliminaries

Unstructured Grid Solver

Application to Euler Equations

Floating Shock-Fitting Implementation

Unstructured Grids Shock-Fitting Results




Problems appear at the end of each chapter.

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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.