Mathematical and Numerical Analysis on Compressible Flow:
P.D. Lax:
Hyperbolic systems of conservation laws and the mathematical theory of
shock waves (1973);
J. Smoller:
Shock waves and reaction-diffusion equations, 2nd ed. (1994);
E. Godlewski and P-A Raviart:
Hyperbolic systems of conservation laws (1991);
E. Godlewski and P-A Raviart:
Numerical approximation of hyperbolic systems of conservation laws
(1996);
P.L. Lions:
Mathematical Topics in Fluid Mechanics, v.2. Compressible Models (1996)
B. Gustafsson, H-O Kreiss, J. Oliger:
Time Dependent Problems and Difference Methods (1995)
H.-O. Kreiss and J. Lorenz:
Initial-Boundary Value Problems and the Navier-Stokes equations (1989)
CFD:
Ch. Hirsch:
Numerical computation of internal and external flows,
v.1. Fundamentals of numerical discretization (1988),
v.2. Computational methods for inviscid and viscous flows. (1990);
R.D. Richtmyer and K.W. Morton:
Difference methods for initial-value problems, 2d ed. (1967);
C. Canuto, et al.:
Spectral Methods in Fluid Dynamics (1988)
R. Peyret and T.D. Taylor:
Computational Methods for Fluid Flow (1985)
C.A.J. Fletcher:
Computational Techniques for Fluid Dynamics (1991)
Basic Numerics, Scientific Computing, and
Software Packages:
Numerical Recipes in Fortran or in C, Cambridge University Press
H.-O. Kreiss:
Numerical Methods for Solving Time-Dependent Problems
for Partial Differential Equations (1978)
J.C. Strikwerda:
Finite Difference Schemes and Partial Differential
Equations (1989)
Morton, Mayers:
Numerical Solution of Partial Differential Equations
Scientific Computing
MAPL660, Fall 98,
MAPL661, Spring 99
Computational Fluid Dynamics,
MAPL698b, Fall 99
Financial Modeling & Simulation,
MAPL699, Spring 00
* Derivation of hyperbolic systems and conservation laws;
* The Euler equations and the Navier-Stokes equations of gas dynamics;
* Mach number and the compressibility
* Other Applications:
- Shallow water equations;
- Acoustic waves in the atmosphere, the ocean, or solids;
- Electro-magnetic waves, including visible light, radar;
- Traffic dynamics on a crowded road,;
- Porous media flow, e.g. water or petroleum under the earth
- Combustion of gases involving detonation and deflagration waves;
- Magnetohydrodynamical waves in plasma and ionized gases; etc
Mathematical Theory
* Linear hyperbolic systems:
- characteristics and Riemann invariants,
- solution to the Riemann problem,
- applications to acoustics and elasticity
* Scalar nonlinear conservation laws,
- Burgers' equation and traffic flow models;
- Shock formation and weak solutions,
- rarefaction waves, the Riemann problem;
- Non-uniqueness and entropy conditions,
* Nonlinear systems of hyperbolic equations:
- Solution of the nonlinear Riemann problem;
- Shock waves, rarefaction waves, contact discontinuities.
- shock speed and Rankine-Hugoniot relation
Numerical Methods
* First-order Monotone methods:
- stability and convergence theory, Lax equivalence Theorem
- the CFL condition,
- Lax-Friedriches scheme, upwind methods and Godunov's method,
- kinetic scheme and flux splitting
- numerical flux functions, numerical viscosity and modified equation
* Second-order and high-resolution methods:
- Lax-Wendroff scheme and MacCormack scheme
- flux limiters and slope limiters, PPM and ENO scheme
- total variation diminishing (TVD) methods and FCT scheme
* Boundary conditions: periodic, non-reflecting, solid walls
* Far field boundary conditions
* Approximate Riemann solvers, e.g. Roe solvers for nonlinear systems
* Stability and convergence of nonlinear methods:
- conservative scheme and the Lax-Wendroff theorem,
- total variation stability
* Multi-dimensional problems: dimensional splitting and unsplit methods
More applications (depending in part on the interests of students)
* Curvilinear grids and unstructured grids
* Adaptive mesh refinement
* Source terms, (e.g. chemical reactions, gravitational forces),
fractional step methods
* Multi-gird method for steady state computation
(ASME) The American Society of Mechanics Engineers
International
(AIAA) The American Institute of Aeronautics and Astronautics
(APS) The American Physical Society, Division of Fluid
Dynamics
Yahoo: Fluid Dynamics Index
Fluid Mechanics Related Web Pages
ICASE Fluid Mechanics Highlights