Numerical Analysis II

Instructor:                Gregory  Beylkin

Class hours/location:       MW 3.00-4.15 @ECCR 1B55

Text for APPM 5600/5610

• K. Atkinson, Introduction to Numerical Analysis, Second Edition.

• G. Golub and C. Van Loan, Matrix Computations, Chapters 2-5, 7, 10.
• A. Iserles, A First Course in the Numerical Analysis of Differential Equations
• J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (except Chapter 1 and Sections 4A, 7.7, 8.8-8.10)
• K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations

Syllabus

The Matrix Eigenvalue Problem

• Theoretical preliminaries
  
• Diagonalizable, normal, self-adjoined (Hermitian) matrices
• Eigenvalue problem
• Schur's decomposion, Singular Value Decomposition (SVD)
  
• Eigenvalue location, error analysis, and stability
• The power and inverse power methods
• Householder reflections, Givens rotations, Hessenberg form of a matrix
• QR Iteration
• Algorithms for a self-adjoined (Hermitian) tridiagonal matrix

Numerical Methods for Ordinary Differential Equations

• Existence, uniqueness, and stability theory
• Euler's method
• Linear multistep methods
• Predictor-corrector methods
• Convergence and stability theory for multistep methods
• Stiff ODEs
• Runge-Kutta methods
• Boundary value problems
• *** A fast algorithm for (linear) two-point boundary value problem
• *** An introduction to symplectic integrators

A review: Fourier Integrals, Fourier Series and Fast Fourier Transform (FFT) algorithm

Introduction to Linear PDEs

• Classification of linear PDE's
• Inital value and boundary value problems
• Finite Difference discretization of elliptic PDEs and associated linear algebra problems
• Algorithms for the Poisson equation
• Finite Difference discretization of hyperbolic  PDEs
• Finite Difference discretization of parabolic  PDEs: Crank-Nicolson and ADI methods
• Stability and convergence: CFL condition, von Neumann stability analysis, Lax equivalence theorem
• Pseudospectral methods
• *** A brief introduction to multiresolution methods for the Poisson equation

Introduction to Linear Integral Equations

• *** Integral equations of the potential theory
• *** Discretization of integral equations and associated linear algebra problems
• *** Fast methods for solving integral equations