# Syllabus: APMA 2560

## Syllabus:  APMA 2560

 Lecture 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: Syllabus APMA 2560 One-Dimensional Finite Element Implementation Handout #2 Two-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 grids, Math. 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: Handout #1 Self-Adjoint Operators APMA 2560 Handout #2 First-Derivatives APMA 2560 Handout #3 Gauss-Legrendre quadrature world record APMA 2560 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: Handouts #1 Spectra and eigenspectra (from Canuto et al.) Handout #2 Inverse inequality for Legendre expansions