Numerical Methods for Time-Dependent PDEs
AMSC 612, Spring 2011
Course Information
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.
Examples:
Shallow-water eqs, PDEs from image processing, geometrical optics, incompressible Euler eqs, nonlinear models for traffic flow,...
Assignment #1 [ pdf file ]
Lecture notes: Fourier expansions
[ 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: One-dimensional wave equation
[ notes ] (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 ]
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
Lecture notes: Matrices: Eigenvalues, Norms and Powers
[ pdf file ]
Assignment #2 [ pdf file ]
with selected answers [ pdf file ]
- 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 ]
Lecture notes: two-dimensional wave equation:
[ notes ] simulation
[ movie ]
(J. Balbas)
Assignment #4 [ pdf file ]
with selected answers [ pdf file ]
Related material:
[ CFL paper 1928 ]
Convergence Theory
- Accuracy
- Stability: von Neumann analysis; power-bounded symbols
- Convergence: stability implies convergence (Lax equivalence theorem)
- Numerical dissipation; Kreiss matrix theorem
Related material: From semi-discrete to fully-discrete: stability of Runge-Kutta schemes
[ I ]
[ II ]
Assignment #5
[ pdf file ] with selected answers [ pdf file ]
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
- Difference approximations to initial-boundary value problems
- Eigen-mode analysis: Godunov-Ryabenkii condition
- UKC the resolvent condition and stability
- Examples
Lecture notes: Stability --- power-boundedness vs. the resolvent condition
[ notes ]
Assignment #7
[ pdf file ] with selected answers [ pdf file ]
Spectral Methods
- Fourier and Chebyshev methods
- Spectral accuracy, stability and convergence
Lecture notes: Chebyshev expansions
[ pdf file]
Lecture notes: Fourier & Chebyshev methods for time dependent problems
[ pdf file]
Assignment #8
[ pdf file ] with selected answers [ pdf file ]
Nonlinear Problems
- Nonlinear Conservation laws
- Smooth solutions: linearization. Strang's theorem
- Nonlinear conservation laws: shock wand rarefaction waves
- Viscosity solutions and L1 scalar theory
Lecture notes: Introduction to scalar conservation laws
[ notes ]
- Finite-volume and discontinuous Galerkin methods
- Conservative schemes
- Godunov and Godunov-type schemes: limiters and high-resolution
- Roe, Godunov and Lax-Friedrichs schemes: L1-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 central schemes: 2nd-order TVD scheme and more can be found here
Spectral viscosity
[1989 paper]
Final Assignment #10
[ pdf file ]
- Hamilton-Jacobi equations (extra material - if there is time ....)
- Related nonlinear problems: incompressible Euler eqs (extra material - if there is time ....)
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
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
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