Table Of ContentHussaini . Van Leer· Van Rosendale (Eds.)
Upwind and High-Resolution Schemes
Springer
Berlin
Heidelberg
New York
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo
M. Yousuff Hussaini
Bram van Leer
John Van Rosendale (Eds.)
Upwind
and
High-Resolution
Schemes
Springer
Professor M. Yousuff Hussaini
Program in Computational Science & Engineering
Florida State University
Tallahassee, FL 32306-3075, USA
Professor Bram van Leer
Department of Aerospace Engineering
University of Michigan
Ann Arbor,_MI 48109-2118, USA
Dr. John Van Rosendale
ICASE, NASA Langley Research Center
Mail Stop 132C
Hampton, VA 23681-0001, USA
Library of Congress Cataloging-in-Publication Data.
Upwind and higll-resolution schemes / M. Yousuff Hussaini, Bram van Leer, John Van Rosendale (eds.).
p. cm. Includes bibliographical references. ISBN-13: 978-3-642-6445241 (Berlin: alk. paper)1.Fluid dynamics -
Mathematical models. 2. Compressibility. 1. Hussaini, M. Yousuff. II. Leer, Bram van. III. Van Rosendale,
John R. QA911.U68 1997 532'.0535'0013-dc21 96-39965
ISBN-13: 978-3-642-64452-8 e-ISBN -13: 978-3-642-60543-7
DOl: 10.1 007/978-3-642-60543-7
This work is subject to copyrigllt. All rights are reserved, whether the whole or part of the material is con
cerned, specifically the rigllts of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microf1lrn or in any other way, and storage in data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Viola-
tions are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1997
Softcover reprint of the hardcover 1s t edition 1997
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant protec
tive laws and regulations and therefore free for general use.
Typesetting of the Preface, the two introductory articles, the annotations to the four chapters,
and the bibliography: Data conversion by Springer-Verlag
Cover design: design & production GmbH, Heidelberg
SPIN 10548466 55/3144 - 5 4 3 21 0 - Printed on acid-free paper
This volume is dedicated to
AMI HARTEN
September 22, 1946 - August 5, 1994
Preface
One of the major achievements in computational fluid dynamics has been the de
velopment of numerical methods for simulating compressible flows. A problem
atic feature of compressible flows is the possible occurrence of shocks, structures
much too narrow to be resolved on any practical grid. John von Neumann's so
lution to this problem is almost as old as his concept of the electronic computer.
During World War II, in order to compute the effect of a strong explosion in the
atmosphere, he introduced the concept of artificial viscosity, spreading a shock
over a few computational cells and thereby "capturing" it without identifying it
as a discontinuity. However, it was not until the 1970s that computational sci
entists succeeded in formulating methods that combine higher-order accuracy in
smooth regions with a sharp, oscillation-free representation of embedded shocks
- methods now known as "high-resolution schemes" .
During the early 1980s knowledge of high-resolution schemes permeated aero
nautical science; from then on this discipline took the lead in further advancing
these methods. In this chain of events, leASE played a crucial role. leASE,
where computational scientists and aerospace engineers meet without effort,
proved to be the right place at the right time for interdisciplinary endeavors,
the breeding of new ideas, and technology transfer. The list of leASE reports
from 1980 onward gives testimony to this historic development.
In 1989, one of us (Hussaini) had the idea of capturing the spirit of this
exciting period by reprinting a collection of the most significant papers on high
resolution schemes. Ami Harten, one of the principal contributors to the subject,
was asked to write a technical introduction to this collection. He did begin such
an introduction, but with the many projects he was involved in, never finished
it, and the manuscript was eventually lost. The task was later completed by
another key player in the field, Phil Roe.
The present volume collects in one place many of the most significant pa
pers in the development of high-resolution schemes, as this occurred at leASE.
Some of these stand alone, representing a milestone in the development of high
resolution schemes; others are selected as being typical of a particular thread
of research, without being comprehensive. The papers follow a historical per
spective, and have been grouped by research theme into four chapters, each
beginning with notes on the relevance of the papers included.
The volume begins with a historical introduction by one of us (Van Leer).
Following this comes the technical introduction by Phil Roe to which we have
alluded. While the papers in this volume are self contained, they were written
VII
from the point of view of their day, when knowledge was incomplete. Roe's
introduction describes high-resolution schemes from the modern vantage point,
placing the papers in perspective. At the same time, it connects the historical
development of high-resolution schemes to the modern push towards truly mul
tidimensional schemes, an effort being actively pursued at many places around
the world.
The remainder of the volume consists of four chapters. Chapter I contains
papers highlighting the early development of approximate Riemann solvers in
the context of monotonic schemes, which are necessarily of first-order accu
racy. To obtain second-order accuracy, one has to relax the monotonicity re
quirement to the weaker condition of diminishing of total-variation, and Chap
ter II consists of papers dealing with second-order total-variation-diminishing
(TVD) schemes. To obtain arbitrarily high-order accuracy, the total-variation
diminishing requirement has to be relaxed to boundedness of total-variation.
Chapter III consists of papers which provide a historical development of total
variation-bounded (TVB) schemes at ICASE. Finally, Chapter IV collects to
gether papers pertaining to inherently multidimensional schemes, which preserve
the favorable features of their one-dimensional predecessors.
We hope this volume will find its way onto the bookshelves of those who share
with us a fascination for computational science in general, and computational
fluid dynamics in particular.
Finally, we wish to express our appreciation to Shelly M. Johnson and Shan
non L. Keeter, members of ICASE's excellent secretarial staff, whose patient
hard work and organizational skills were invaluable in putting this volume to
gether. We would also like to thank Professor W. Beiglbock for encouraging this
effort and for seeing the volume through to completion.
M. Yousuff Hussaini
Bram van Leer
John Van Rosendale
VIII
Contents
ICASE and the History of High-Resolution Schemes
B. van Leer .............................................. 1
A Brief Introduction to High-Resolution Schemes
P. Roe. . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . .. 9
Chapter I. Riemann Solvers and Upwind Schemes
Annotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
1. On the Relation Between the Upwind-Differencing Schemes
of Godunov, Engquist-Osher and Roe
B. van Leer .............................................. 33
2. On Upstre!J,m Differencing and Godunov-Type Schemes
for Hyperbolic Conservation Laws
A. Harten, P.D. Lax, and B. van Leer. . . . . . . . . . . . . . . . . . . . . . . . .. 53
3. Flux-Vector Splitting for the Euler Equation
B. van Leer .............................................. 80
Chapter II. Second-Order Total-Variation Diminishing Schemes
Annotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93
1. A Comparative Study of Computational Methods
in Cosmic Gas Dynamics
G.D. van Albada, B. van Leer, and W. W. Roberts, Jr. . . . . . . . . . . . . .. 95
2. High Resolution Schemes and the Entropy Condition
S. Osher and S. Chakravarthy ................................ 104
3. Convergence of Generalized MUSCL Schemes
S. Osher ................................................. 134
4. A Simplified TVD Finite Difference Scheme Via Artificial Viscosity
S.F. Davis ............................................... 149
5. On Central-Difference and Upwind Schemes
R. C. Swanson and E. Turkel ................................. 167
IX
Chapter III. Essentially Non-oscillatory Schemes
Annotation ................................................. 185
1. Uniformly High-Order Accurate Nonoscillatory Schemes. I
A. Harten and S. Osher ..................................... 187
2. Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III
A. Harten, B. Engquist, S. Osher, and S.R. Ghakmvarthy ........... 218
3. ENO Schemes with Subcell Resolution
A. Harten ................................................ 291
4. Efficient Implementation of Essentially Non-oscillatory
Shock-Capturing Schemes, II
G.-W. Shu and S. Osher ..................................... 328
5. Essentially Nonoscillatory Spectral Fourier Methods
for Shock Wave Calculations
W. Gai, D. Gottlieb, and G.-W. Shu ........................... 375
6. Cell Averaging Chebyshev Methods for Hyperbolic Problems
W. Gai, D. Gottlieb, and A. Harten ............................ 397
Chapter IV. Multidimensional Schemes
Annotation ................................................. 421
1. A Rotationally Biased Upwind Difference Scheme
for the Euler Equations
S.P. Davis ............................................... 423
2. Discrete Models for the Numerical Analysis of Time-Dependent
Multidimensional Gas Dynamics
P.L. Roe ................................................. 451
3. Discontinuous Solutions to Hyperbolic Systems
Under Operator Splitting
P.L. Roe ................................................. 470
4. High Resolution Finite Volume Methods on Arbitrary Grids
Via Wave Propagation
R.J. Le Veque ............................................. 491
5. Design of an Essentially Nonoscillatory Reconstruction Procedure
on Finite-Element-Type Meshes
R. Abgrall ................................................ 519
6. A Contribution to the Great Riemann Solver Debate
J. J. Quirk ................................................ 550
7. An Approximate Riemann Solver for Magnetohydrodynamics
(That Works in More Than One Dimension)
K.G. Powell .............................................. 570
Papers Reprinted .......................................... 585
x
leASE
and the History of High-Resolution Schemes
Bram van Leer
The need for high-resolution schemes is a direct consequence of the nonlinear
properties of hyperbolic systems of conservation laws such as the Euler equa
tions of inviscid compressible flow. Due to the nonlinearity of these equations,
any compression wave, no matter how smooth initially, will evolve into a shock
discontinuity, while the interaction of shock waves in turn may create shear and
entropy discontinuities. When attempting to compute a solution with such fea
tures on a grid of discrete meshes, we have the choice of representing these as
actual discontinuities - as in "shock fitting" [1, 2J, "jump recovery" [3J, or "sub
cell resolution" [4], or spreading them artificially, as in "shock capturing" [5, 6J.
In the latter technique the goal is to automatically represent any discontinuity
by a sharp but oscillation-free transition, with most of the jump realized within
one to three meshes, without compromising accuracy away from such jumps.
The term "high resolution" appeared first in an article by A. Harten [7J, who
attributed it to P. R. Woodward.
For an appreciation of the problem of designing a high-resolution scheme for
the Euler equations it is useful to first consider modeling the linear advection of
a step function. Here we immediately run into a famous theorem presented by
S. K. Godunov [8J in his influential 1959 paper: if an advection scheme preserves
the monotonicity of the solution it is at most first-order accurate. Thus, if
our step representation is to be monotone, it will necessarily be quite smeared.
This result could discourage anyone attempting to improve advection schemsj
fortunately, there is a way to circumvent it. In the proof of this theorem it is
tacitly assumed that the linear advection equation is approximated by a linear
discretizationj once nonlinear discretizations are admitted the theorem no longer
stands and high-resolution schemes become possible.
In the same publication Godunov points out that, among the simplest ad
vection schemes, the first-order upwind scheme is in some sense "best": it is the
most accurate monotonicity-preserving scheme. He then shows how to extend
the concept of upwind differencing to a nonlinear system of conservation laws,
viz, the one-dimensional Lagrangean flow equations, by using the solution of
Riemann's initial-value problem. This is an evolution problem for a hyperbolic
conservation law, with initial values
Uo(x) = {Ul' x < 0,
Ur, x> 0.
In Godunov's method the numerical solution at each time level is regarded to
M. Y. Hussaini et al. (eds.), Upwind and High-Resolution Schemes 1
© Springer-Verlag Berlin Heidelberg 1997
