MA59800, Fall 2020

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

Instructor: Jie Shen

TTh 3:00p-4:15p at LWSN B155 (or with lectures online via zoom ID: 2128080179, password: 6666)


All in person class (unless announced otherwise)!


Office: MATH 450
Online Office Hours: Tu 10:30-11:45; F 9:45-11:00
(zoom ID: 2128080179, password: 6666) or by appointment 
Phone: 4-1923
Message: 4-1901
E-mail: shen7@purdue.edu

Projects

Downloadable programs

Shenfun: a high performence package based on spectral-Galerkin methods


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:


Topics:

    PART I.
  • Some fundamental tools in numerical PDEs
  • Fourier-spectral methods
  • basic results for polynomial approximations
  • Galerkin and collocation methods for elliptic PDEs

    PART II.

  • Time discretization schemes for gradient systems
  • 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 phase-field models

A review paper on SAV methods for gradient flows