Course 8023 - Numerical Differential Equations I - Fall 2016

Official Information
Course Number:Mathematics 8023.001
Course Title:Numerical Differential Equations I
Time:MW 9:00-10:20
Place:617 Wachman Hall
Instructor: Benjamin Seibold
Instructor Office:518 Wachman Hall
Instructor Email: seibold(at) 
Office Hours:MW 10:30-11:30am
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.
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, Runge-Kutta 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:Runge-Kutta, 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
   Read: ODE, IVP, BVP
08/31/2016   Lec 2
Partial differential equations, linear vs. nonlinear, well-posedness
   Read: PDE, Well-posed problem
09/07/2016   Lec 3
Fourier methods for linear PDE IVPs, Laplace/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, Runge-Kutta methods
09/21/2016   Lec 7
General Runge-Kutta methods, ERK-DIRK-IRK, order conditions
09/26/2016   Lec 8
Embedded methods and adaptive time stepping
   Read: Adaptive stepsize
09/28/2016   Lec 9
Linear multistep methods
10/03/2016   Lec 10
Zero-stability and convergence
   Read: Zero-stability
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
10/17/2016   Lec 14
Boundary value problems, consistency
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
Advection-diffusion-reaction problems, variable coefficient diffusion, anisotropic diffusion
11/02/2016   Lec 18
Singularly perturbed problems, weak derivatives
11/04/2016   Lec 19
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, Lax-equivalence theorem
11/14/2016   Lec 22
Von Neumann stability analysis, multi-dimensional problems
11/16/2016   Lec 23
V. Wave propagation: advection, MOL, Lax-Wendroff, Lax-Friedrichs
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, Langborg-Hansen, 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
Runge-Kutta 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 n-th 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
Set-up 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 one-way 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, Lax-Friedrichs, Lax-Wendroff, and Crank-Nicolson. For each method, the corresponding growth factor for von Neumann stability analysis is shown.
mit18086_fd_transport_limiter.m Nonlinear finite differences for the one-way 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, Lax-Friedrichs and Lax-Wendroff as linear methods, and as a nonlinear method Lax-Wendroff-upwind 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.
  • Fawzi Belguet: Introducing chaos: evaluating Lorenz equations
  • Abhijit Biswas: Jet schemes with nonlinear interpolants
  • Chia-Han Chou: Numerical solutions for Black-Scholes equation using time-stepping methods
  • Joshua Finkelstein: Numerical integration schemes for molecular dynamics
  • Stephan Grein: Anisotropic diffusion filtering
  • Amit Harel: A time-dependent wave equation and its relationship with non-linear elasticity
  • Shanlin Jin: Image zooming based on a numerical differential equations approach
  • Mikkel Langborg-Hansen: Simulation and optimization of winning strategies in a boat race
  • Zhi Li: Numerical methods for the Sethi model and its application in multi-product situation
  • Rabie Ramadan: Solving a Stokes problem via CLAWPACK solutions of a hyperbolic balance law system
  • Najhem Salehi: Spectral methods for non-linear models for long waves in the ocean