Official Information  

Course Number:  Mathematics 8024.001 
Course Title:  Numerical Differential Equations II 
Time:  TR 11:0012:20 
Place:  527 Wachman Hall 
Instructor:  Benjamin Seibold 
Instructor Office:  518 Wachman Hall 
Instructor Email:  seibold(at)temple.edu 
Office Hours:  T 12:2013:30, R 10:0011:00 
Official:  Course Syllabus 
Course Textbook: 
There is no single textbook for the course. The material is inspired by the following resources (in decreasing order of relevance):

Grading Policy  
The final grade consists of three parts, each counting 33.3%:  
Homework Problems:  Each homework assignment will be worked on for two weeks. 
Course project:  From the third until the second to last week, each participant works on a course project. Students can/should suggest projects themselves. The instructor is quite flexible with project topics as long as they are new, interesting, and relate to the topics of this course. The project grade involves a midterm report (20%) and a final report (50%), and a final presentation (30%). Due dates are announced in class. 
Exams:  05/08/2017 
Outline  
This course is designed for graduate students of all areas who are interested in numerical methods for differential equations, with focus on a rigorous mathematical basis. Many modern and efficient approaches are presented, after fundamentals of numerical approximation are established. This course continues last semester's 8023. Topics covered include nonlinear hyperbolic conservation laws, finite volume methods, ENO/WENO, SSP RungeKutta schemes, wave equations, spectral methods, interface problems, level set method, HamiltonJacobi equations, Stokes problem, NavierStokes equation, and pseudospectral approaches for fluid flow. Further topics possible upon request.  
Course Schedule  
01/17/2017 Lec 1  Review of 8023

01/19/2017 Lec 2  I. Hyperbolic conservation laws:
derivation, examples, weak solutions
Read:
Conservation law,
Hyperbolic PDE

01/24/2017 Lec 3  Riemann problem, shocks, entropy solution
Read:
Riemann problem,
Shock

01/26/2017 Lec 4  Finite difference methods for discontinuous solutions

01/31/2017 Lec 5  Failure of finite different methods for nonlinear problems

02/02/2017 Lec 6  Consistent finite difference schemes, finite volume methods
Read:
Finite volume method

02/07/2017 Lec 7  Godunov's methods, nonconvex flux functions
Read:
Godunov's method

02/14/2017 Lec 8  Nonlinear stability theory

02/16/2017 Lec 9  Highorder methods, limiters
Read:
Flux limiter,
Godunov's theorem

02/17/2017 Lec 10  Linear hyperbolic systems
Read:
Examples of equations

02/21/2017 Lec 11  Nonlinear hyperbolic systems, approximate Riemann solvers

02/23/2017 Lec 12  Higher dimensions, semidiscrete methods

02/24/2017 Lec 13  SSP time stepping, ENO/WENO

03/07/2017 Lec 14  Operator splitting, stiff source terms, staggered grids
Read:
Strang splitting,
Arakawa grids

03/09/2017 Lec 15  II. Interface problems: Front propagation

03/21/2017 Lec 16  Numerical interface representation

03/23/2017 Lec 17  HamiltonJacobi equations

03/24/2017 Lec 18  Numerical schemes for HJ equations

03/30/2017 Lec 19  III. Spectral methods: periodic problems

04/04/2017 Lec 20  Fast Fourier transform
Read:
FFT

04/06/2017 Lec 21  Nonperiodic problems

04/07/2017 Lec 22  Applications

04/11/2017 Lec 23  IV. Fluid flows: Calculus of variations, Stokes problem

04/13/2017 Lec 24  Saddle point problems, staggered grid approaches

04/14/2017 Lec 25  NavierStokes equations
Read:
NavierStokes equations

04/18/2017 Lec 26  Semispectral methods for the NavierStokes equations

04/25/2017 Lec 27  Project presentations

04/27/2017 Lec 28  Project presentations

05/08/2017  Final Examination 
Matlab Programs  
Chapter I: Hyperbolic conservation laws  
mit18086_fd_transport_growth.m 
Finite differences for the oneway wave equation, additionally plots
von Neumann growth factor Approximates solution to u_t=u_x, which is a pulse travelling to the left. The methods of choice are upwind, downwind, centered, LaxFriedrichs, LaxWendroff, and CrankNicolson. For each method, the corresponding growth factor for von Neumann stability analysis is shown. 
mit18086_fd_transport_limiter.m 
Nonlinear finite differences for the oneway wave equation with
discontinuous initial conditions Solves u_t+cu_x=0 by finite difference methods. Of interest are discontinuous initial conditions. The methods of choice are upwind, LaxFriedrichs and LaxWendroff as linear methods, and as a nonlinear method LaxWendroffupwind with van Leer and Superbee flux limiter. 
temple8024_godunov_shallow_water.m 
Godunov's method for the shallow water equations Godunov's method for the shallow water equations, using an approximate Riemann solver with Roe averaging. No entropy fix is implemented. 
temple8024_weno_claw.m 
WENO finite volume code for onedimensional scalar conservation laws Solves u_t+f(u)_x = 0 by a semidiscrete approach, in which 5th order WENO is used for the reconstruction of the Riemann states at cell boundaries, and the 3rd order SSP ShuOsher scheme is used for the time stepping. 
mit18086_fd_waveeqn.m 
Finite differences for the wave equation Solves the wave equation u_tt=u_xx by the Leapfrog method. The example has a fixed end on the left, and a loose end on the right. 
Chapter II: Interface problems  
mit18086_levelset_front.m 
Level set method for front propagation under a given front velocity field First order accurate level set method with reinitialization to compute the movement of fronts in normal direction under a given velocity. 
Chapter III: Spectral methods (all direct links to Nick Trefethen's codes)  
cheb.m  
p4.m  Periodic spectral differentiation using matrices 
p5.m  Periodic spectral differentiation using FFT 
p13.m  Solving a linear BVP 
p14.m  Solving a nonlinear BVP 
p15.m  Solving an eigenvalue problem 
p16.m  Solving the 2D Poisson equation 
p17.m  Solving the 2D Helmholtz equation 
Chapter IV: Fluid flows  
mit18086_navierstokes.m 
Finite differences for the incompressible NavierStokes equations in a box Solves the incompressible NavierStokes equations in a rectangular domain with prescribed velocities along the boundary. The standard setup solves a lid driven cavity problem. This Matlab code is compact and fast, and can be modified for more general fluid computations. You can download a Documentation for the program. 
Homework Problem Sets  
 
Course Projects  
Project proposals due: January 31, 2017.
Project midterm reports due: March 23, 2017. Project final reports due: May 1, 2017.
