Numerical Methods in Partial Differential Equations

AMSC 612, Fall 2005

Course Information

Lecture4122 CSIC Bldg. #406; TuTh 14-15:15pm
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   Email:
Office HoursBy appointment
Midterm Date TBA, 4122 CSIC Bldg. #406
Final Date TBA, 4122 CSIC Bldg. #406
Grading50% Homework + midterm, 50% Final

Course Description

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.

  1. Initial Value Problems

    1. Initial value problems of hyperbolic type
      • The wave equation - the energy method and Fourier analysis 
      • Weak and strong hyperbolicity -- systems with constant coefficients
      • Hyperbolic systems with variable coefficients
    2. Initial value problems of parabolic type
      • The heat equation -- Fourier analysis and the energy method
      • Parabolic systems
    3. Well-posed problems
      Lecture notes: Time dependent problems [ pdf file ]
      Assignment #1 [ pdf file ]

  2. Finite Difference Approximations for Initial Value Problems

    • Preliminaries
      • Discretization. grid functions, their Fourier representation, divided differences.
      • Upwind and centered Euler's schemes: accuracy, stability, CFL and convergence
    • Transport equations: examples of finite difference schemes
      • Lax-Friedrichs schemes: numerical dissipativity
      • Lax-Wendroff schemes: second- (and higher-) order accuracy
      • Leap-Frog scheme: the unitary case
      • Crank-Nicolson scheme: implicit schemes
    • Forward Euler for the heat equation
  3. Convergence theory
    • Accuracy
    • von Neumann analysis: the symbol
    • Stability and power-boundedness
    • Kreiss matrix theorem; numerical dissipation
    Lecture notes:  Power-boundedness [ HTML] [ pdf file ]

    Related material:  From semi-discrete to fully-discrete: stability of Runge-Kutta schemes by the energy method [ I ] [ II ]
    Assignment #2 [ pdf file ]
  4. Problems with variable coefficients
    • Strong stability and freezing coefficients
    • Symmetrizers and the Lax-Nirenberg result
    • Dissipative schemes
    • The energy method: positive schemes, skew-symmetric differencing
  5. Multi-dimensional problems
    • ADI and splitting methods
  6. Nonlinear problems
    • Linearization: Strang's theorem
    • Nonlinear conservation laws, Hamilton-Jacobi eq's, ...

    Midterm [ pdf file ]


  7. Initial Boundary Value Problems and their Approximation

    1. One dimensional hyperbolic systems
      • Method of characteristics
      • The energy method: maximal dissipativity
    2. Multi-dimensional hyperbolic systems
      • Eigen-mode analysis
      • Uniform Kreiss Condition (UKC) and resolvent stability
    3. Difference approximations to initial-boundary value problems
      • Eigen-mode analysis: eth Godunov-Ryabenkii condition
      • UKC the resolvent condition and stability
      • Examples

  8. Elliptic Problems and Finite Difference Approximations

    1. Finite difference discretizations
    2. Iterative solution of large sparse systems
    3. Multigrid methods

  9. Spectral Methods

    1. Fourier and Chebyshev methods
    2. Spectral accuracy, stability and convergence
      Lecture notes: spectral methods for time dependent problems [ pdf file]
    FINAL  [ pdf file ]

    Epilogue - have you been paying attention in your numerical analysis course?


    References

    F. John, PDEs, Applied Mathematical Sciences 1, 4th ed., Springer-Verlag, 1982

    R. Richtmyer and B. Morton,  Difference Methods for Initial-Value Problems, 2nd ed., Interscience, Wiley, 1967

    H.-O. Kreiss and J. Oliger,  Discrete Methods for Time Dependent Problems

    B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time Dependent Problems and Difference Methods, Wiley, 1995

    V. Thomee,  Stability Theory for PDEs, SIAM Rev. 11 (1969)

    R. J. LeVeque, Numerical Methods for Conservation Laws, 2nd ed., Birkhäuser Verlag, 1992

    Eitan Tadmor