Official Information  

Course Number:  Mathematics 8023.001 
Course Title:  Numerical Differential Equations I 
Time:  MW 9:0010:20 
Place:  617 Wachman Hall 
Instructor:  Benjamin Seibold 
Instructor Office:  518 Wachman Hall 
Instructor Email:  seibold(at)temple.edu 
Office Hours:  MW 10:3011:30am 
Official:  Course Syllabus 
Course Textbook:  Randall J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations  Steady State and Time Dependent Problems, SIAM, 2007 
Further Reading: 
L.C. Evans,
Partial Differential Equations,
Graduate Studies in Mathematics, V. 19, American Mathematical Society, 1998
L.N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000 Randall J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002 
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:  December 15, 2016. 
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. Of particular focus are a qualitative understanding of the considered ordinary and partial differential equation, RungeKutta and multistep methods, fundamentals of finite difference, finite volume, and finite element methods, and important concepts such as stability, convergence, and error analysis.  
Fundamentals:  Stability, convergence, error analysis, Fourier approaches. 
Methods:  RungeKutta, multistep, finite difference, finite volume, finite element methods. 
Problems:  Dynamical systems, boundary value problems, Poisson equation, advection, diffusion, wave propagation, conservation laws. 
Course Schedule  
08/29/2016 Lec 1  I. Fundamental Concepts: Ordinary differential equations, IVPs vs. BVPs

08/31/2016 Lec 2  Partial differential equations, linear vs. nonlinear, wellposedness
Read:
PDE,
Wellposed problem

09/07/2016 Lec 3  Fourier methods for linear PDE IVPs, Laplace/Poisson equation
Read:
Fourier series,
Poisson equation

09/08/2016 Lec 4  Heat equation, transport/wave equation

09/14/2016 Lec 5  II. Numerical methods for ODE IVPs:
fundamentals, truncation errors

09/19/2016 Lec 6  Taylor series methods, RungeKutta methods

09/21/2016 Lec 7  General RungeKutta methods, ERKDIRKIRK, order conditions

09/26/2016 Lec 8  Embedded methods and adaptive time stepping
Read:
Adaptive stepsize

09/28/2016 Lec 9  Linear multistep methods
Read:
Linear multistep method

10/03/2016 Lec 10  Zerostability and convergence
Read:
Zerostability

10/05/2016 Lec 11  Absolute stability
Read:
Stability

10/10/2016 Lec 12  Stiff problems
Read:
Stiff equation

10/12/2016 Lec 13  III. Finite difference methods for BVPs:
generalized finite difference approach
Read:
Finite difference method

10/17/2016 Lec 14  Boundary value problems, consistency
Read:
Boundary value problem

10/19/2016 Lec 15  Convergence and stability for BVPs, Neumann b.c.

10/24/2016 Lec 16  2D Poisson equation, deferred correction

10/26/2016 Lec 17  Advectiondiffusionreaction problems, variable coefficient diffusion, anisotropic diffusion

11/02/2016 Lec 18  Singularly perturbed problems, weak derivatives

11/04/2016 Lec 19  Finite element method
Read:
Finite element method

11/07/2016 Lec 20  IV. Parabolic equations:
heat equation, method of lines, stability
Read:
Method of lines

11/09/2016 Lec 21  Accuracy, Laxequivalence theorem
Read:
Lax equivalence theorem

11/14/2016 Lec 22  Von Neumann stability analysis, multidimensional problems

11/16/2016 Lec 23  V. Wave propagation:
advection, MOL, LaxWendroff, LaxFriedrichs
Read:
LaxWendroff,
LaxFriedrichs

11/28/2016 Lec 24  Upwind methods, stability analysis
Read:
Upwind

11/30/2016 Lec 25  Modified equation
Read:
Modified equation

12/05/2016 Lec 26  Linear hyperbolic systems, wave equation, outlook
Read:
Hyperbolic PDE

12/07/2016 Lec 27  Project presentations:
Belguet, Biswas, Chou, Finkelstein, Grein, Harel

12/12/2016 Lec 28  Project presentations:
Jin, LangborgHansen, Li, Ramadan, Salehi

12/15/2016  Final Examination 
Matlab Programs  
Chapter I: Fundamental concepts  
mit18086_linpde_fourier.m 
Four linear PDE solved by Fourier series Shows the solution to the IVPs u_t=u_x, u_t=u_xx, u_t=u_xxx, and u_t=u_xxxx, with periodic b.c., computed using Fourier series. The initial condition is given by its Fourier coefficients. In the example a box function is approximated. 
Chapter II: Numerical methods for ODE initial value problems  
ode1.m ode2.m ode3.m ode4.m ode5.m 
RungeKutta methods of orders 1,2,3,4, and 5 Initial value problem ODE are solved approximated equidistant time steps. 
Chapter III: Finite difference methods for boundary value problems  
mit18086_stencil_stability.m 
Compute stencil approximating a derivative given a set of points and plot
von Neumann growth factor Computes the stencil weights which approximate the nth derivative for a given set of points. Also plots the von Neumann growth factor of an explicit time step method (with Courant number r), solving the initial value problem u_t = u_nx. Example for third derivative of four points to the left: >> mit18086_stencil_stability(3:0,3,.1) 
mit18086_poisson.m 
Numerical solution of Poisson equation in 1D, 2D, and 3D Setup of system matrices using Matlab's kron. 
mit18336_poisson1d_error.m 
Perform numerical error analysis for the Poisson equation A differentiable but oscillatory right hand side is considered. 
Chapter IV: Parabolic equations  
mit18086_fd_heateqn.m 
Finite differences for the heat equation Solves the heat equation u_t=u_xx with Dirichlet (left) and Neumann (right) boundary conditions. 
Chapter V: Wave propagation  
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. 
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. 
Homework Problem Sets  
 
Course Projects  
Project proposals due: September 16, 2016.
Project midterm reports due: October 24, 2016. Project final reports due: December 12, 2016.
