AMSC/CMSC 661: Scientific Computing II

HW collection (HW 1 - 11)

Final Exam

Numerical methods  for Elliptic PDEs

- Linear elliptic equations. Modeling using elliptic PDEs. Existence and Uniqueness theorems, weak and strong maximum principles.

- Finite difference methods in 2D: different types of boundary conditions, convergence.

- Variational and weak formulations for elliptic PDEs.

- Finite element method in 2D

Codes: elpot.m, MyFEMCat.m, cat.png

Refs:

Numerical Linear Algebra for Sparse Matrices

- Basic iterative methods: Jacobi, Gauss-Seidel, SOR.

- Multigrid.

- Krylov subspace methods: the conjugate gradient and generalizations

Refs:

 J. Nocedal and S. Wrigth, Numerical Optimization, 2nd edition, Springer  (see Chapter 5 for Conjugate Gradient methods)

Numerical Methods for Time-Dependent PDEs

- Parabolic equations:

* Heat equation. Finite difference methods: explicit and implicit. Basic facts about stability and convergence.

* Solving heat equation in 2D using finite element method.

* Method of lines. An example of a nonlinear equation (the Boussinesq equation).

- Linear advection equation:

* Finite difference methods. Basic facts about stability and convergence. The CFL condition.

* Fourier transform. Dispersion analysis. Phase and group velocities.

- Hyperbolic conservation laws:

* Shock speed and the Rankine-Hugoriot condition, weak solutions, entropy condition and vanishing viscosity solution

* Numerical methods for conservation laws: conservative form, consistency, Godunov's and Glimm's methods.

Codes:  cat.png

Refs:

 Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 9, Chapter 10, Appendix E

 Fourier Transform links: TrefethenP. Cheung

Fourier and Wavelet Transform Methods

- Continuous and Discrete Fourier transforms

- Spectral methods for solving linear and nonlinear PDEs

- The fast Fourier transform

- Nyquist frequency, sampling theorem

- Continuous and discrete wavelet transforms

- Haar and Daubechies wavelets, approximation properties, fast wavelet transforms

- Application of wavelets to image processing

Codes: linear_dispersion.m, linear_dispersion0

Codes:

  solution of u_t +u_{xxx} = 0  for x \in [0, 2pi] using DFT and exact time integration

 linear_dispersion01.m: solution of u_t +u_{xxx} = 0 for x \in [0, 1] using DFT and exact time integration

 solution of the Korteweg - de Vries equation using DFT and a method involving DFT and exact time integration of the linear part of the RHS

   image compression using wavelets. Input image: Lenna.png available at https://en.wikipedia.org/wiki/File:Lenna.png (Links to an external site

Refs:

 Stephane Mallet, A wavelet tour of signal processing. The sparse way. 3rd edition. Academic Press, Elsevier, 2009

 Lecture notes on wavelets and multi resolution analysis:

Phillip K. Poon (U. of Arizona),

Brani Vidakovich  (GATech),

Copyright 2010, 2015 , 2017, 2018  by Maria Cameron