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Tables of Contents for High-Order Methods for Computational Physics
Chapter/Section Title
Page #
Page Count
High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes
1
68
Remi Abgrall, Universite Bordeaux I, 351 Cours de la Liberation 33 405 Talence Cedex
Thomas Sonar, Institut fur Angewandte Mathematik, Universitat Hamburg BundesstraBe 55, D-20146 Hamburg, Germany
Oliver Friedrich, Institut fur Angewandte Mathematik, Universitat Hamburg BundesstraBe 55, D-20146 Hamburg, Germany
Germain Billet, ONERA, DSNA, Avenue de la Division Leclerc, 92 230 Chatillon sous Bagneux, France
1 Introduction
2
2
2 An overview of finite volume schemes
4
2
2.1 The Euler equations
4
1
2.2 Finite volume formulation
5
1
3 The reconstruction step: classical methods
6
4
3.1 An essentially non oscillatory Lagrange interpolation
6
2
3.2 Application to finite volume schemes
8
1
3.3 Possible extensions to higher dimensions and their weaknesses
9
1
4 The reconstruction problem on unstructured meshes
10
6
4.1 Preliminaries
10
1
4.2 Some general results about problem P
11
1
4.3 The case of a nonsmooth function
12
2
4.4 Three polynomial expansions
14
2
5 The explicit calculation of the reconstruction: Muhlbach expansions, Tschebyscheff systems and divided differences
16
6
5.1 A linear system for divided differences
18
1
5.2 A recurrence relation for divided differences
18
2
5.3 Some examples
20
2
6 The ENO reconstruction
22
2
6.1 ENO on general meshes
22
1
6.2 Numerical examples
22
2
7 Weighted ENO reconstruction
24
3
7.1 Motivation of WENO reconstruction
24
1
7.2 Choice of weights
25
1
7.3 Required modifications of the reconstruction algorithm
26
1
7.4 A stencil selection algorithm that does not need triangles
26
1
8 Other recovery techniques
27
1
9 A class of high order numerical schemes for compressible flow simulations
28
2
9.1 Numerical tests
29
1
9.2 Some remarks on the formal accuracy of the scheme
29
1
10 Multiresolution Analysis
30
5
10.1 Introduction
30
1
10.2 Harten's multiresolution analysis on general meshes
31
3
10.3 Numerical examples
34
1
10.4 Multiresolution analysis and ENO schemes
34
1
10.5 A numerical experiment
34
1
11 Other applications: Hamilton Jacobi equations
35
2
12 Schemes with adaptive limiters and fluxes
37
12
12.1 MUSCL Approach and Flux Splittings
38
2
12.2 First-order Error Terms in Space
40
3
12.3 Second-order Error Terms in Space
43
3
12.4 Multi-time stepping algorithm
46
1
12.5 Applications
47
1
12.6 Summary
48
1
13 Conclusion
49
13
A Details of (36)
62
3
A.1 Continuity equation
62
2
A.2 Momentum equation
64
1
A.3 Energy equation
65
1
B A piece of code
65
4
Discontinuous Galerkin Methods for Convection-Dominated Problems
69
156
Bernardo Cockburn
Adaptive Spectral Element Methods for Turbulence and Transition
225
100
Ronald D. Henderson
hp-FEM for Fluid Flow Simulation
325
114
Christoph Schwab
High Order ENO and WENO Schemes for Computational Fluid Dynamics
439
Chi-Wang Shu