Elsevier S & T

Godunov-type Schemes: An Introduction for Engineers

Guinot, V.

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Table of contents
  • Cover
  • Contentsxi
  • Prefacev
  • Acknowledgementsix
  • Notationxvii
  • VariabIesxvii
  • Operatorsxxiii
  • Subscripts and superscriptsxxiii
  • Othersxxiv
  • Chapter 1. Scalar conservation laws1
  • 1.1 Definitions and basic notions1
  • 1.2 The Riemann problem18
  • 1.3 A linear conservation law: the advection equation25
  • 1.4 A convex conservation law: the Burgers equation29
  • 1.5 A concave conservation law: the LWR model32
  • 1.6 A non-convex conservation law: the Buckley-Leverett equation38
  • 1.7 Extension to multiple dimensions46
  • Chapter 2. Hyperbolic systems of conservation laws51
  • 2.1 Definitions52
  • 2.2 A linear system: the water hammer equations62
  • 2.3 Two-phase flow in pipes68
  • 2.4 A 2x2 model for traffic flow73
  • 2.5 The open channel flow equations with solute transport82
  • 2.6 The shallow water equations in two dimensions89
  • Chapter 3. An outline of Godunov-type schemes93
  • 3.1 The six steps of Godunov-type algorithms94
  • 3.2 Lagrangian schemes103
  • 3.3 Multidimensional problems105
  • 3.4 Stability constraints114
  • Chapter 4. The Godunov method for scalar laws in one dimension117
  • 4.1 The linear advection equation118
  • 4.2 Application to the inviscid Burgers equation125
  • 4.3 Application to the LWR model137
  • 4.4 Application to the Buckley-Leverett equation146
  • Chapter 5. The Godunov method for systems of conservation laws155
  • 5.1 Application to the water hammer equations155
  • 5.2 Application to the simplified model for two-phase flow in pipes172
  • 5.3 Application to a 2x2 traffic flow model187
  • 5.4 Application to the open channel flow equations200
  • Chapter 6. Higher-order schemes225
  • 6.1 Principle of higher-order schemes226
  • 6.2 The MUSCUPLM schemes244
  • 6.3 The PPM scheme252
  • 6.4 The DPM scheme265
  • 6.5 Boundary conditions for higher-order schemes274
  • 6.6 Application example278
  • Chapter 7. Multidimensional schemes291
  • 7.1 Multidimensional hyperbolic systems of conservation laws292
  • 7.2 Alternate directions304
  • 7.3 The finite volume approach309
  • 7.4 Wave splitting318
  • 7.5 Computational examples334
  • 7.6 Higher-order multidimensional schemes342
  • Chapter 8. Large-time-step algorithms345
  • 8.1 Front tracking algorithms347
  • 8.2 Implicit/explicit methods356
  • 8.3 The time-line reconstruction method366
  • 8.4 Computational examples382
  • Chapter 9. Concluding remarks387
  • Appendix A. Notions in mathematics389
  • A.1 Linear algebra389
  • A.2 Accuracy/consistency, stability, convergence400
  • Appendix B. Riemann solvers415
  • B.1 Exact Riemann solvers415
  • B.2 The HLL Riemann solver417
  • B.3 Roe’s Riemann solver420
  • B.4 Approximate-state solvers427
  • Appendix C. Sample codes431
  • C.1 The code Linadv431
  • C.2 The code ‘Burgers’435
  • C.3 The code ‘LWR’438
  • C.4 The code ‘BL’442
  • C.5 The code ‘WatHam’445
  • C.6 The code ‘2phase’448
  • C.7 The code ‘Traffic’452
  • C.8 The code ‘Channel’456
  • C.9 The code ‘Sh2D’461
  • C.10 The code ‘Large’466
  • References471
  • Index481
Book details
  • Vendor Elsevier S & T
  • SKU 9780444511553
  • ISBN-13 9780080532585
  • Author Guinot, V.
  • Category Technology & Engineering
  • Subject Hydraulics

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Description eBook delivery & access Reviews
Godunov-type schemes appear as good candidates for the next generation of commercial modelling software packages, the capability of which to handle discontinuous solution will be a basic requirement. It is in the interest of practising engineers and developers to be familiar with the specific features of discontinuous wave propagation problems and to be aware of the possibilities offered by Godunov-type schemes for their solution.


This book aims to present the principles of such schemes in a way that is easily understandable to practising engineers.
The features of hyperbolic conservation laws and their solutions are presented in the first two chapters. The principles of Godunov-type schemes are outlined in a third chapter. Chapters 4 and 5 cover the application of the original Godunov scheme to scalar laws and to hyperbolic systems of conservation laws respectively. Chapter 6 is devoted to higher-order schemes in one dimension of space. The design of such a scheme is described for the general case and applied to some well-known schemes such as the MUSCL and PPM schemes. Chapter 7 focuses on multidimensional problems. The classical alternate directions and finite volume approaches are presented together with the wave splitting technique that is described in depth with an application to two-dimensional systems. Chapter 8 deals with large-time step algorithms. These include front tracking-based methods, explicit-implicit techniques and the time-line interpolation technique. Three appendices provide notions on accuracy and stability issues, Riemann solvers and the user instructions for the computational codes provided in the enclosed CD-ROM.