These notes were developed for a graduate-level course on the theory and numerical solution of nonlinear hyperbolic systems of conservation laws. Part I deals with the basic mathematical theory of the equations: the notion of weak solutions, entropy conditions, and a detailed description of the wave structure of solutions to the Riemann problem. The emphasis is on tools and techniques that are indispensable in developing good numerical methods for discontinuous solutions. Part II is devoted to the development of high resolution shock-capturing methods, including the theory of total variation diminishing (TVD) methods and the use of limiter functions. The book is intended for a wide audience, and will be of use both to numerical analysts and to computational researchers in a variety of applications.
Author(s): Randall J. LeVeque, R. Leveque
Publisher: Birkhauser
Year: 1992
Language: English
Pages: 228
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
1.1 Conservation laws......Page 14
1.2 Applications......Page 15
1.3 Mathematical difficulties......Page 21
1.4 Numerical difficulties......Page 22
1.5 Some references......Page 25
2.1 Integral and differential forms......Page 27
2.2 Scalar equations......Page 29
2.3 Diffusion......Page 30
3.1 The linear advection equation......Page 32
3.1.1 Domain of dependence......Page 33
3.1.2 Nonsmooth data......Page 34
3.2 Burgers' equation......Page 36
3.3 Shock formation......Page 38
3.4 Weak solutions......Page 40
3.5 The Riemann Problem......Page 41
3.6 Shock speed......Page 44
3.7 Manipulating conservation laws......Page 47
3.8 Entropy conditions......Page 49
3.8.1 Entropy functions......Page 50
4.1 Traffic flow......Page 54
4.1.1 Characteristics and "sound speed"......Page 57
4.2 Two phase flow......Page 61
5.1 The Euler equations......Page 64
5.1.1 Ideal gas......Page 66
5.1.2 Entropy......Page 67
5.2 Isentropic flow......Page 68
5.4 The shallow water equations......Page 69
6.1 Characteristic variables......Page 71
6.3 The wave equation......Page 73
6.4 Linearization of nonlinear systems......Page 74
6.4.1 Sound waves......Page 76
6.5 The Riemann Problem......Page 77
6.5.1 The phase plane......Page 80
7.1 The Hugoniot locus......Page 83
7.2 Solution of the Riemann problem......Page 86
7.3 Genuine nonlinearity......Page 88
7.4 The Lax entropy condition......Page 89
7.5 Linear degeneracy......Page 91
7.6 The Riemann problem......Page 92
8.1 Integral curves......Page 94
8.2 Rarefaction waves......Page 95
8.3 General solution of the Riemann problem......Page 99
8.4 Shock collisions......Page 101
9.1 Contact discontinuities......Page 102
9.2 Solution to the Riemann problem......Page 104
II Numerical Methods......Page 108
10 Numerical Methods for Linear Equations......Page 110
10.1 The global error and convergence......Page 115
10.2 Norms......Page 116
10.3 Local truncation error......Page 117
10.4 Stability......Page 119
10.5 The Lax Equivalence Theorem......Page 120
10.6 The CFL condition......Page 123
10.7 Upwind methods......Page 125
11 Computing Discontinuous Solutions......Page 127
11.1 Modified equations......Page 130
11.1.1 First order methods and diffusion......Page 131
11.1.2 Second order methods and dispersion......Page 132
11.2 Accuracy......Page 134
12 Conservative Methods for Nonlinear Problems......Page 135
12.1 Conservative methods......Page 137
12.2 Consistency......Page 139
12.3 Discrete conservation......Page 141
12.4 The Lax-Wendroff Theorem......Page 142
12.5 The entropy condition......Page 146
13 Godunov's Method......Page 149
13.1 The Courant-Isaacson-Reel method......Page 150
13.2 Godunov's method......Page 151
13.3 Linear systems......Page 153
13.4 The entropy condition......Page 155
13.5 Scalar conservation laws......Page 156
14 Approximate Riemann Solvers......Page 159
14.1 General theory......Page 160
14.1.1 The entropy condition......Page 161
14.2 Roe's approximate Riemann solver......Page 162
14.2.1 The numerical flux function for Roe's solver......Page 163
14.2.2 A sonic entropy fix......Page 164
14.2.3 The scalar case......Page 166
14.2.4 A Roe matrix for isothermal flow......Page 169
15.1 Convergence notions......Page 171
15.2 Compactness......Page 172
15.3 Total variation stability......Page 175
15.5 Monotonicity preserving methods......Page 178
15.6 11-contracting numerical methods......Page 179
15.7 Monotone methods......Page 182
16.1 Artificial Viscosity......Page 186
16.2 Flux-limiter methods......Page 189
16.2.1 Linear systems......Page 195
16.3 Slope-limiter methods......Page 196
16.3.1 Linear Systems......Page 200
16.3.2 Nonlinear scalar equations......Page 201
16.3.3 Nonlinear Systems......Page 204
17.1 Evolution equations for the cell averages......Page 206
17.2 Spatial accuracy......Page 208
17.3 Reconstruction by primitive functions......Page 209
17.4 ENO schemes......Page 211
18 Multidimensional Problems......Page 213
18.1 Semi-discrete methods......Page 214
18.2 Splitting methods......Page 215
18.4 Multidimensional approaches......Page 219
Bibliography......Page 221
