Jian-Guo Liu : Math 226 Numerical Methods for Partial Differential Equations

Class hours: MWF 1:30-2:20 Physics 119
Office hours: MWF 11:00-12:00 Physics 295, Tel: 660-2841
Email: jliu "at " math.duke.edu, http://www.math.umd.edu/~jliu"

Course Description:
Duke Online Course Synopsis Handbook:
Numerical solution of hyperbolic conservation laws. Conservative difference schemes, modified equation analysis and Fourier analysis, Lax-Wendroff process. Gas dynamics and Riemann problems. Upwind schemes for hyperbolic systems. Nonlinear stability, monotonicity and entropy; TVD, MUSCL, and ENO schemes for scalar laws. Approximate Riemann solvers and schemes for hyperbolic systems. Multidimensional schemes. Adaptive mesh refinement. Finite difference and spectral methods for time-dependent problems. Hyperbolic and parabolic systems, initial and initial-boundary value problems of hyperbolic and parabolic types. Stability and convergence theories for linear and nonlinear problems. Finite difference methods for elliptic problems.



Numerical methods for PDE:
Lloyd N. Trefethen: Spectral Methods in MATLAB (Software, Environments, Tools), SIAM, 2001
Stig Larsson and Vidar Thomee: Partial Differential Equations with Numerical Methods (Texts in Applied Mathematics), Springer-Verlag, 2005
William L. Briggs, Van Emden Henson, and Steve F. McCormick: A Multigrid Tutorial, SIAM, 2000
Bengt Fornberg: A Practical Guide to Pseudospectral Methods (Cambridge Monographs on Applied and Computational Mathematics), 1998
Alfio Quarteroni and Alberto Valli: Numerical Approximation of Partial Differential Equations (Springer Series in Computational Mathematics), 1997
K. Eriksson, D. Estep, P. Hansbo, and C. Johnson: Computational Differential Equations, Cambridge University Press, 1996

Lawrence C. Evans: Partial Differential Equations (Graduate Studies in Mathematics, V. 19) 1998
Walter A. Strauss: Partial Differential Equations: An Introduction, Wiley, 1992
Fritz John: Partial Differential Equations (Applied Mathematical Sciences), Springer; 4th ed., 1991

Compressible Flow:
R. Courant and K.O. Friedriches: Supersonic Flow and Shock Waves(1948);
J.D. Anderson: Modern compressible flow with historical perspective, 2nd ed., (1990);
W.G. Vincenti and C.H. Kruger: Introduction to physical gas dynamics (1965);
L. Landau and E. Lifschitz: Fluid Mechanics (1959);
D.J. Acheson: Elementary Fluid Dynamics (1990).
A.J. Chorin and J.E. Marsden: A Mathematical Introduction to Fluid Mechanics
G. B. Whitham: Linear and nonlinear waves (1974);
M. Van Dyke: An Album of Fluid Motion (1982);

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)

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

My Papers on CFD


Introduction to Compressible Flow
    * 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

Web Sites Links:

Fluid Mechanics:
(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

Flow Image Collections:
Fluid Dynamics Picture of the Week (contains many other links to fluid dynamics topic areas).
National University of Singapore-- T.T. Lim's home page. Spectacular images of fluid flows, including a