Official Information  

Course Number:  Mathematics 8024.001 
Course Title:  Numerical Differential Equations II 
Time:  TR 10:3011:50 
Place:  527 Wachman Hall 
Instructor:  Benjamin Seibold 
Instructor Office:  518 Wachman Hall 
Instructor Email:  seibold(at)temple.edu 
Office Hours:  TR 2:003:00pm 
Official:  Course Syllabus 
Course Textbooks: 
Randall J. LeVeque,
Finite Volume Methods for
Hyperbolic Problems, Cambridge University Press, 2002
Stanley Osher, Ron Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer, 2002 Jan S. Hesthaven, Tim Warburton, Nodal Discontinuous Galerkin Methods, Springer, 2008 
Further Reading: 
Randall J. LeVeque,
Finite Difference Methods for
Ordinary and Partial Differential Equations  Steady State and Time Dependent Problems
SIAM, 2007
L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, V. 19, American Mathematical Society, 1998 
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%), a final report (50%), and a final presentation (30%). Due dates are announced in class. 
Exams:  May 2, 2022. 
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, interface problems, level set method, HamiltonJacobi equations, discontinuous Galerkin methods, Stokes problem, NavierStokes equation, and pseudospectral approaches for fluid flow. Further topics possible upon request.  
Course Schedule  
01/18/2022 Lec 1  Review of 8023

01/20/2022 Lec 2  I. Hyperbolic conservation laws:
derivation, examples
Read:
Conservation law,
Hyperbolic PDE

01/25/2022 Lec 3  Weak solutions, Riemann problem, shocks
Read:
Riemann problem,
Shock

01/27/2022 Lec 4  Entropy, finite difference methods for discontinuous solutions

02/01/2022 Lec 5  Failure of finite different methods for nonlinear problems

02/03/2022 Lec 6  Consistent and conservative finite difference schemes
Read:
Finite volume method

02/08/2022 Lec 7  Finite volume methods, Godunov's method
Read:
Godunov's method

02/10/2022 Lec 8  Nonconvex flux functions

02/15/2022 Lec 9  Nonlinear stability theory

02/17/2022 Lec 10  Highorder methods
Read:
Godunov's theorem

02/18/2022 Lec 11  Slope and flux limiters
Read:
Flux limiter

02/22/2022 Lec 12  Linear hyperbolic systems
Read:
Examples of equations

02/24/2022 Lec 13  Nonlinear hyperbolic systems
Read:
Examples of equations

02/25/2022 Lec 14  Approximate Riemann solvers, higher dimensions

03/08/2022 Lec 15  II. Other approaches for transport:
semidiscrete methods, SSP time stepping
Read:
MUSCL scheme

03/10/2022 Lec 16  ENO/WENO, operator splitting
Read:
Strang splitting

03/15/2022 Lec 17  Stiff source terms, staggered grids
Read:
Arakawa grids

03/17/2022 Lec 18  Discontinuous Galerkin methods

03/22/2022 Lec 19  Discontinuous Galerkin methods

03/29/2022 Lec 20  III. Interface problems: Front propagation

03/31/2022 Lec 21  Numerical interface representation, level set method

04/01/2022 Lec 22  HamiltonJacobi equations, numerical methods

04/05/2022 Lec 23  IV. Fluid flows: Calculus of variations, Stokes problem

04/07/2022 Lec 24  Saddle point problems, staggered grid approaches

04/12/2022 Lec 25  NavierStokes equations
Read:
NavierStokes equations

04/14/2022 Lec 26  Projection methods, staggered grid methods
Read:
Projection method

04/19/2022 Lec 27  Semispectral methods for the NavierStokes equations

04/28/2020 Lec 28  Project presentations

05/02/2020  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. 
Chapter II: Other approaches for transport  
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. 
dgclaw_compute.m dgclaw_shallowwater.m 
Discontinuous Galerkin method for hyperbolic conservation laws. Solver file for RKDG method with limiting in characteristic variables (dgclaw_compute.m), with an example of the shallow water equations (dgclaw_shallowwater.m). 
Chapter III: 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 IV: Fluid flows  
mit18086_navierstokes.m 
Finite differences for the incompressible NavierStokes equations in a 2d 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. 
mit18336_spectral_ns2d.m 
Spectral method for incompressible NavierStokes in a periodic 2d box Solves the incompressible NavierStokes equations in a rectangular domain with periodic boundary conditions, using a semispectral method and the fast Fourier transform. (Code 2008 by JeanChristophe Nave) 
Additional Course Materials  
 
Homework Problem Sets  
 
Course Projects  
Project proposals due: January 20, 2022.
Project midterm reports due: March 10, 2022. Project final reports due: May 3, 2022.
