MA69200, Fall 2018

Topics in Spectral Methods, Computational Fluid Dynamics and Computational Materials Science

Instructor: Jie Shen

TTh 9:00-10:15 at REC 113

Office: MATH 450
Office Hours: Tu 10:30-11:45; F 9:45-11:00
or by appointment 
Phone: 4-1923
Message: 4-1901


Downloadable programs

Course outline:

This is a course on selected topics on spectral methods for solving PDEs and on time discretization schemes for computational fluid dynamics and computational materials science, including:


    PART I.
  • Some fundamental tools in numerical PDEs
  • Fourier-spectral methods
  • basic results for polynomial approximations
  • Galerkin and collocation methods for elliptic PDEs
  • Basic introduction to Fractional PDEs
  • Spectral methods using generalized Jacobi functions for fractional PDEs

    PART II.

  • Time discretization schemes for gradient flows
  • Time discretization schemes for Navier-Stokes equations
  • Phase-field models for multi-phase incompressible flows and their time discretization

Prerequisite: A good knowledge of calculus, linear algebra, numerical analysis and some basic programming skills are essential. Some knowledge of real analysis and functional analysis will be helpful but not necessary.

Requirement: There will be no exam. Course grades will be based on homework assignments and programming projects.

Reference materials:

J. Shen, T. Tang and L. Wang, "Spectral Methods: Algorithms, Analysis and Applications" (Springer Series in Computational Mathematics, V. 41, Springer, Aug. 2011), and the associated Matlab codes

L. N. Trefethen, Spectral Methods in Matlab, SIAM 2000

A review paper on spectral methods for fractional PDEs

A review paper on phase-field models

A review paper on SAV methods for gradient flows