Syllabus

Syllabus:  APMA 2560

Lecture

Content and reading assignments

1     

Introduction to variational calculus – Examples 

Theory of the first (Gateaux) variation and the Euler-Lagrange equations 

Theory of the second (Gateaux) variation and Legendre’s necessary condition 

Variational problems with constraints • Boundary Conditions – Examples 

Reading:

Introduction to the Calculus of Variations, by Hans Sagan (Dover).

Methods of Mathematical Physics, vol. 1, Courant & Hilbert, Chapter IV.

2

Methods of Weighted Residuals

Collocation method 

Least-squares method 

Galerkin and Petro-Galerkin method 

Tau method 

Multi-domain formulation - Finite Elements 

Reading:

Karniadakis and Sherwin, sections 2.1-2.2

Gottlieb and Orszag, section 2

Handouts:

One-dimensional Finite Element Implementation

Tw0-dimensional Finite Element Implementation

3

Abstract formulation of FEM for elliptic problems 

One dimensional discontinuous FEM

Reading:

C. Johnson's book:  Chapter 2

Paper by Zhang and Shu

4

Examples of simple PDEs and their physical properties

Fornberg's finite difference method for computing derivatives.

Phase error analysis and high-order method; efficiency.

Infinite-order finite difference formula.

Reading:

B. Fornberg, Generation of high order finite differences on arbitrarily spaced gridsMath. Comput: 51:699-706, 1988.

Korteweg-de Vries equation

5

Fourier Spectral Methods

Fourier-Galerkin method; stability for hyperbolic problems 

Fourier-Collocation method; stability for hyperbolic problems 

Stability of parabolic equations 

Stability of nonlinear equations 

Reading:

Section 6 of Gottlieb and  Orszag

Chapter 3 of Hesthaven et al.

6

Approximation Theory and Resolution Rules-of-Thumb 

Chebyshev polynomials and convergence 

Regular versus singular Sturm-Liuville problems 

Legendre polynomials and convergence 

Laguerre and Hermite polynomials 

Askey family of orthogonal polynomials 

Handouts:

Self-Adjoint Operators

First-Derivatives

Gauss-Legreendre quadrature world record

Reading:

Gottlieb & Orszag, section 3

Karniadakis & Sherwin, Appendix A

7

Polynomial Spectral Methods 

Galerkin formulation; stability 

Collocation formulation; stability 

Tau and penalty methods 

Reading:

Sections 2-7-8 of Gottlieb & Orszag

Chapters 7-8 of Hesthaven et al.

8

Spectral Methods for Non-Smooth Problems

Reading:

Chapter 9 from Hesthaven et al.

9

Finite Elements in Two-Dimensions:  Theory and Implementation

Reading:

Chapter 4 from C. Johnson

Handout of Lecture 2

10

Time discretization - Spectral Methods 

Stability of continuous and semi-Discrete problems 

An example of a fully discrete system (Euler/Legendre collocation) 

Eigenspectra of derivative operators - von Neuman stability 

Common time-stepping methods (multi-step, multi-stage, integrating factos, and strong-preserving stability) 

Reading:

Read chapter 10 of Hesthaven et al.

Handouts:

Spectra and eigenspectra of first derivatives (from Canuto et al.)

Inverse inequality for Legendre expansions