Maria K. Cameron

AMSC/CMSC 661: Scientific Computing II

HW collection (HW 1 - 11)

Final exams: 2016, 2017, 2018, 2022

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.


• [3] 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.m, multigrid.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 Conjucate 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: Trefethen, P. Cheung


• [4] Method of stationary phase


• [5] Cameron's notes on Burger'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:

• [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.m: 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


• [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)

Codes:

• [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


• [3] Cameron’s note on the Korteweg - de Vries equation


• [4] 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)