Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for SourcesThis book is devoted to finite volume methods for hyperbolic systems of conservation laws. It differs from previous expositions on the subject in that the accent is put on the development of tools and the design of schemes for which one can rigorously prove nonlinear stability properties. Sufficient conditions for a scheme to preserve an invariant domain or to satisfy discrete entropy inequalities are systematically exposed, with analysis of suitable CFL conditions. |
Contents
Quasilinear systems and conservation laws | 1 |
12 Conservative systems | 2 |
13 Invariant domains | 4 |
14 Entropy | 5 |
15 Riemann invariants contact discontinuities | 9 |
Conservative schemes | 13 |
21 Consistency | 14 |
22 Stability | 15 |
42 Consistency | 71 |
43 Stability | 74 |
44 Required properties for Saint Venant schemes | 75 |
45 Explicitly wellbalanced schemes | 77 |
46 Approximate Riemann solvers | 79 |
461 Exact solver | 81 |
462 Simple solvers | 82 |
47 Suliciu relaxation solver | 83 |
222 Entropy inequalities | 16 |
23 Approximate Riemann solver of Harten Lax Van Leer | 19 |
231 Simple solvers | 22 |
232 Roe solver | 24 |
234 Vacuum | 25 |
24 Relaxation solvers | 26 |
241 Nonlocal approach | 29 |
242 Rusanov flux | 30 |
243 HLL flux | 32 |
244 Suliciu relaxation system | 33 |
245 Suliciu relaxation adapted to vacuum | 36 |
246 Suliciu relaxationHLLC solver for full gas dynamics | 40 |
25 Kinetic solvers | 45 |
251 Kinetic solver for isentropic gas dynamics | 47 |
26 VFRoe method | 48 |
27 Passive transport | 50 |
28 Secondorder extension | 53 |
281 Secondorder accuracy in time | 58 |
Source terms | 65 |
31 Invariant domains and entropy | 66 |
32 Saint Venant system | 67 |
Nonconservative schemes | 69 |
41 Wellbalancing | 70 |
48 Kinetic solver | 84 |
49 VFRoe solver | 85 |
410 Fwave decomposition method | 87 |
411 Hydrostatic reconstruction scheme | 88 |
4111 Saint Venant problem with variable pressure | 93 |
4112 Nozzle problem | 94 |
412 Additional source terms | 96 |
4121 Saint Venant problem with Coulomb friction | 97 |
413 Secondorder extension | 99 |
4131 Secondorder accuracy | 100 |
4132 Wellbalancing | 103 |
4133 Centered flux | 104 |
4134 Reconstruction operator | 105 |
Multidimensional finite volumes with sources | 107 |
51 Nonconservative finite volumes | 108 |
52 Wellbalancing | 109 |
54 Additional source terms | 112 |
55 Twodimensional Saint Venant system | 113 |
Numerical tests with source | 117 |
127 | |
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Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation ... François Bouchut No preview available - 2005 |