MATH3475 & MATH5476M Course Overview

• Teaching
  

  The 15-credit level-3 course (2.5-hour exam) consists of 27 lectures and 6 examples classes at a rate of 3 contact hours in each week of the second semester. The teaching is supported by 6 examples sheets, 29 printed handouts and 14 Maple worksheets, which can be obtained here.

  The 20-credit level-5 course (3-hour exam) augments the above with 11 additional contact hours, comprising revision of essential material (section ii. here) of the prerequisite module MATH3474 in weeks I-V, and an advanced topic (varying from session to session) in weeks VI-XI; an additional examples sheet will accompany this topic. Note that the level-5 examination will contain a compulsory question on the advanced topic.
  
  

• General content
  

  The course will consider more advanced and/or recent numerical methods for solving BVPs and IVPs in 1- and 2-D. It has the following broad content: a tabular résumé of lecture times and titles, including detailed lecture-by-lecture information, can be downloaded in PDF format.

  • Boundary-Value Problems (14 lectures): Strong, weak & variational formulations; self-adjoint 2nd-order differential operators; 2-point BVPs; inhomogeneous BCs; basis & trial functions; defect/residual; essential & natural BCs; Galerkin weighted-residual method; application to general and self-adjoint BVPs; pointwise residual; nodal collocation; regular- & Chebyshev-node spectral collocation; variational principles; Euler-Lagrange equation; energy functionals; Rayleigh-Ritz method & relationship to Galerkin method; inhomogeneous mixed BCs; Rayleigh-Ritz method for self-adjoint 2-D BVPs; finite element basis functions; linear & quadratic Lagrangian elements; cubic Hermitian elements; errors in and convergence of the finite-element method; Gauss-Legendre quadrature.
    
    
  • Initial-Value Problems (10 lectures): Stability of single- and multi-step methods for ODEs; linearisation; reduced equation; parasitic solutions; strong, weak, relative & absolute stability; stability polynomial; complex reduced equation; hk stability diagram; parabolic & hyperbolic PDEs; stability of 1-D advection equation; truncation error; consistency; equivalent PDE; rounding error; convergence; Lax' equivalence theorem; eigenvalue stability analysis; normal matrices; Fourier/von Neumann error decomposition; error resolution; aliasing; amplification factor; CFL condition; dissipation & dispersion; decay of high frequencies; spectral error analysis; dissipation & dispersion errors; leading & lagging errors.
    
    
  • Spectral Methods (5 lectures): Trigonometric interpolation; discrete Fourier transform; periodicity; aliasing explained; spectral convergence; spectral differentiation; interpolation on infinite non-periodic domains; Whittaker's cardinal function; trigonometric cardinal functions; spectral differentiation matrices; supergeometric convergence; non-periodic cardinal functions; spectral Chebyshev interpolation on [-1,1]; spectral Chebyshev differentiation matrix.
    
    
  • Integral-Equation Methods (6 lectures, level-5, commencing in week VI): Volterra and Fredholm equations of the first, second and third kinds; characteristic values; quadrature rules for marching methods for 1-D VIE1s ans VIE2s; trial-function approximations for FIE2s; exact and approximate solutions; iterative Neumann-series method for 1-D FIE2s; converged Neumann series; direct methods; Gauss-Legendre quadrature for FIE2s; Nyström method for FIE2s; error analysis.
    
    

Page created by Professor M A Kelmanson on January 23rd 2008; last updated February 23rd 2012.