Numerical Methods for Time-Dependent PDEs
AMSC 612, Fall 2015
Course Information
Place
4122 CSIC Bldg. #406 (unless otherwise stated)
Time
Tuesdays & Thursdays, 14-15:15pm
Tuesday and Thursday Nov. 3-5 meetings, 14-15:15pm: class at CSIC 4141
Additional meetings: Wednesdays 10-12pm on Sep. 30, Nov. 18 and Dec. 2
Instructor Professor Eitan Tadmor
Contact tel.: x5-0648 Email:
eitan Tadmor
Office Hours By appointment
Midterm Thursday, October 15 4122 CSIC Bldg. #406
Final Due: Fri. Dec 11, 2pm 4145 CSIC Bldg. #406
Grading 50% Homework + midterm, 50% Final

Course Description
Time-dependent Partial Differential Equations (PDEs) of hyperbolic and parabolic type. Initial and initial-boundary value problems. Finite difference and spectral methods for time-dependent problems.
The linear theory: accuracy, stability and convergence.
Nonlinear problems: shock discontinuities, viscosity and entropy. Finite volume and discontinuous Galerkin numerical methods.

Introduction
Examples of nonlinear conservation laws: Euler and Navier-Stokes equtions, the shallow-water eqs. PDEs from image processing (examples (i) and (ii) ), geometrical optics, incompressible Euler eqs, nonlinear models for traffic flow,...
Fron nonlinear to constant-coeffiecinets: linearization, method of characteristics
Assignment #1 [ pdf file ]
with selected answers [ pdf file ]
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
Lecture notes on the one-dimensional wave equation
(J. Balbas)
Initial value problems of parabolic type
The heat equation -- Fourier analysis and the energy method
Parabolic systems
Lecture notes: Time dependent problems
[ pdf file ]
Lecture notes: Matrices: Eigenvalues, Norms and Powers
[ 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
Assignment #2
[ pdf file ]
with selected answers [ pdf file ]
Related material:
[ CFL paper 1928 ]
Canonical 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
Assignment #3
[ pdf file ]
with selected answers [ pdf file ]
Assignment #4
[ pdf file ]
with selected answers [ pdf file ]
Convergence Theory
Accuracy
Stability: von Neumann analysis; power-bounded symbols
Convergence: stability implies convergence (Lax equivalence theorem)
Numerical dissipation; Kreiss matrix theorem
Assignment #5 (midterm)
[ pdf file ]
with selected answers [ pdf file ]
Lecture notes: Stability --- power-boundedness vs. the resolvent condition
[ notes ]
Related material: On the stability of the two-dimensional Lax-Wendroff scheme [ pdf file ]
Related material: From semi-discrete to fully-discrete --- stability of Runge-Kutta schemes
[ I ]
[ II ]
Approximations of 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
Multi-Dimensional Problems
ADI and splitting methods
Assignment #6
[ pdf file ]
with selected answers [ pdf file ]
Initial-Boundary Value Problems
One dimensional hyperbolic systems
Method of characteristics
The energy method: maximal dissipativity
Multi-dimensional hyperbolic systems
Eigen-mode analysis
Resolvent stability
A primer on normal mode analysis
Difference approximations to initial-boundary value problems
Eigen-mode analysis: Godunov-Ryabenkii condition
UKC the resolvent condition and stability
Examples
Lecture notes: Stability Theory of Difference Approximations for Mixed initial boundary value problems. II
Gustafsson, Kreiss & Sundstrom, 1972.
[ pdf file ]
Lecture notes(*): L^{2} vs. resolvant stability estimates for mixed systems
[ pdf file ]
Assignment #7
[ pdf file ] with selected answers [ pdf file ]
Spectral Methods
Fourier and Chebyshev methods
Spectral accuracy, stability and convergence
Lecture notes: Fourier expansions, circulant matrices and Fourier differencing
[ pdf file ]
Lecture notes: Chebyshev expansions
[ pdf file ]
Lecture notes: Fourier & Chebyshev methods for time dependent problems
[ pdf file ]
Assignment #8
[ pdf file ]
Nonlinear Problems
Nonlinear Conservation laws
Smooth solutions: linearization. Strang's theorem
Nonlinear conservation laws: shock wand rarefaction waves
Viscosity solutions and L^{1} scalar theory
Lecture notes: Introduction to scalar conservation laws; traffic model
Finite-volume and discontinuous Galerkin methods
Conservative schemes
Godunov and Godunov-type schemes: limiters and high-resolution
Roe, Godunov and Lax-Friedrichs schemes: L^{1} -contraction and TVD property
Lecture notes: Numerical solution of Burgers & Buckley-Levertt equations
[ notes ]
Assignment #9
[ pdf file ]
TVD and non-oscillatory reconstructions
Godunov and Godunov-type schemes: limiters and high-resolution
Entropy stability
Discontinuous Galerkin (DG) method
Related material: Limiters and high-resolution
[ Randy LeVeque notes ] and [Bjorn Sjogreen notes ]
High-resolution schemes: [2nd-order TVD scheme ];
[2nd-order central scheme ] more can be found [here ]
Spectral viscosity
[1989 paper ]
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, 2nd edition, Wiley, 2013
J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth & Brroks, 1989.
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