1st Edition

A Shock-Fitting Primer

By Manuel D. Salas Copyright 2010
    416 Pages 170 B/W Illustrations
    by CRC Press

    416 Pages 170 B/W Illustrations
    by CRC Press

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    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.



    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.


    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.