Wave Propagation in Fluids – Models and Numerical Techniques

Vincent Guinot is professor of hydrodynamic modeling at the University of Montpellier, France. He teaches fluid mechanics, hydraulics, numerical methods and hydrodynamic modeling.

Introduction. Chapter 1. Scalar Hyperbolic Conservation Laws in One Dimension of Space. 1.1. Definitions. 1.2. Determination of the solution. 1.3. A linear law: the advection equation. 1.4. A convex law: the inviscid Burgers equation. 1.5. Another convex law: the kinematic wave for free-surface hydraulics. 1.6. A non-convex conservation law: the Buckley-Leverett equation. 1.7. Advection with adsorption/desorption. 1.8. Conclusions. Chapter 2. Hyperbolic Systems of Conservation Laws in One Dimension of Space. 2.1. Definitions. 2.2. Determination of the solution. 2.3. Specific case: compressible flows. 2.4. A 2x2 linear system: the water hammer equations. 2.5. A nonlinear 2x2 system: the Saint Venant equations. 2.6. A nonlinear 3x3 system: the Euler equations. 2.7. Summary of Chapter 2. Chapter 3. Weak Solutions and their Properties. 3.1. Appearance of discontinuous solutions. 3.2. Classification of waves. 3.3. Simple waves. 3.4. Weak solutions and their properties. 3.5. Summary. Chapter 4. The Riemann Problem. 4.1. Definitions - solution properties. 4.2. Solution for scalar conservation laws. 4.3. Solution for hyperbolic systems of conservation laws. 4.4. Summary. Chapter 5. Multidimensional Hyperbolic Systems. 5.1. Definitions. 5.2. Derivation from conservation principles. 5.3. Solution properties. 5.4. Application to two-dimensional free-surface flow. 5.5. Summary. Chapter 6. Finite Difference Methods for Hyperbolic Systems. 6.1. Discretization of time and space. 6.2. The method of characteristics (MOC). 6.3. Upwind schemes for scalar laws. 6.4. The Preissmann scheme. 6.5. Centered schemes. 6.6. TVD schemes. 6.7. The flux splitting technique. 6.8. Conservative discretizations: Roe's matrix. 6.9. Multidimensional problems. 6.10. Summary. Chapter 7. Finite Volume Methods for Hyperbolic Systems. 7.1. Principle. 7.2. Godunov's scheme. 7.3. Higher-order Godunov-type schemes. 7.4. Summary. Appendix A. Linear Algebra. A.1. Definitions. A.2. Operations on matrices and vectors. A.3. Differential operations using matrices and vectors. A.4. Eigenvalues, eigenvectors. A.4.1. Definitions. A.4.2. Example. Appendix B. Numerical Analysis. B.1. Consistency. B.2. Stability. B.3. Convergence. Appendix C. Approximate Riemann Solvers. C.1. HLL and HLLC solvers. C.2. Roe's solver. Appendix D. Summary of the Formulae. References. Index.