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:

[1] Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 3

[2] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Second Edition, Cambridge University Press, 2005, Chapter 6

[3] Cameron's lecture notes 

[4] S. Larsson and V. Thomee, Partial Differential Equations with Numerical Methods

[5] Jochen Alberty, Carsten Carstensen and Stefan A. Funken, "Remarks around 50 lines of Matlab: short finite element implementation", Numerical Algorithms 20 (1999) 117–137  

 

Numerical Linear Algebra for Sparse Matrices

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

- Multigrid.   

- Krylov subspace methods: the conjugate gradient and generalizations

Codes: basic_iterative_methods.m, Smoothers4multigrid.m, spectra.mmultigrid.m

Refs:

[1] Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM 2007, ( freely available online via the UMD library), Chapter 4

[2] K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Second Edition, Cambridge University Press, 2005, Chapter 7

[3] Yousef Saad, Iterative Methods for Sparse Linear System, SIAM 2003 (see chapter 13 for Multigrid) 

[4] 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: MyFEMheat.m,   cat.png  advection.m

 Refs:

[1] 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

[2] Jochen Alberty, Carsten Carstensen and Stefan A. Funken, "Remarks around 50 lines of Matlab: short finite element implementation", Numerical Algorithms 20 (1999) 117–137

[3] Fourier Transform links: TrefethenP. Cheung 

[4] Method of stationary phase 

[5] Cameron's notes on Burgers's equation

 

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:

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

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

[3] KdVrkm1.msolution of the Korteweg - de Vries equation using DFT and a method involving DFT and exact time integration of the linear part of the RHS

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

Refs:

[1] John P. Boyd, Chebyshev and Fourier Spectral methods, 2nd edition, Dover Publication, Inc., Mineola, New York, 2001

[2] Cameron's notes on Fourier spectral methods

     Cameron’s note on the Korteweg - de Vries equation

[3] Ingrid Daubechies, Ten lectures on wavelets, 1992

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

[5] Lecture notes on wavelets and multi resolution analysis:

Phillip K. Poon (U. of Arizona), 

Brani Vidakovich  (GATech), 

Vlad Balan and Cosmin Condea  (USC)

[6] Sonja Grgic, Kresimir Kers, Mislav Grgic, Image Compression Using Wavelets, IEEE, 0-7803-5662-4/99 (1999)


 Copyright 2010, 2015 , 2017, 2018  by Maria Cameron