Numerical Methods in Partial Differential Equations
AMSC 612, Fall 2005
Course Information
Course Description
Finite difference and spectral methods for timedependent problems.
Hyperbolic and parabolic systems, initial and initialboundary value problems of hyperbolic and parabolic types.
Stability and convergence theories for linear and nonlinear problems.
Finite difference methods for elliptic problems.

Initial Value Problems
 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
 Initial value problems of parabolic type
 The heat equation  Fourier analysis and the energy method
 Parabolic systems
 Wellposed problems
Lecture notes: Time dependent problems [ pdf file ]
Assignment #1 [
pdf file ]
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
 LaxFriedrichs schemes: numerical dissipativity
 LaxWendroff schemes: second (and higher) order accuracy
 LeapFrog scheme: the unitary case
 CrankNicolson scheme: implicit schemes
 Forward Euler for the heat equation
 Convergence theory
 Accuracy
 von Neumann analysis: the symbol
 Stability and powerboundedness
 Kreiss matrix theorem; numerical dissipation
Lecture notes: Powerboundedness
[ HTML]
[ pdf file ]
Related material: From semidiscrete to fullydiscrete: stability of RungeKutta schemes by the energy method
[ I
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[ II
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Assignment #2
[ pdf file ]
 Problems with variable coefficients
 Strong stability and freezing coefficients
 Symmetrizers and the LaxNirenberg result
 Dissipative schemes
 The energy method: positive schemes, skewsymmetric differencing
 Multidimensional problems
 ADI and splitting methods
 Nonlinear problems
 Linearization: Strang's theorem
 Nonlinear conservation laws, HamiltonJacobi eq's, ...
Initial Boundary Value Problems and their Approximation

One dimensional hyperbolic systems
 Method of characteristics
 The energy method: maximal dissipativity
 Multidimensional hyperbolic systems
 Eigenmode analysis
 Uniform Kreiss Condition (UKC) and resolvent
stability

Difference approximations to initialboundary value problems
 Eigenmode analysis: eth GodunovRyabenkii condition
 UKC the resolvent condition and stability
 Examples
Assignment #3 [ pdf file ]
Elliptic Problems and Finite Difference Approximations
 Finite difference discretizations
 Iterative solution of large sparse systems
 Multigrid methods
Spectral Methods
 Fourier and Chebyshev methods
 Spectral accuracy, stability and convergence
Lecture notes: spectral methods for time dependent problems
[ pdf file]
References
F. John, PDEs, Applied Mathematical Sciences 1, 4th ed., SpringerVerlag, 1982
R. Richtmyer and B. Morton,
Difference Methods for InitialValue 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


