| Official Information | |
|---|---|
| Course Number: | Mathematics 8024.001 |
| Course Title: | Numerical Differential Equations II |
| Time: | TR 10:30-11:50 |
| Place: | 527 Wachman Hall |
| Instructor: | Benjamin Seibold |
| Instructor Office: | 518 Wachman Hall |
| Instructor Email: | seibold(at)temple.edu |
| Office Hours: | TR 2:00-3:00pm |
| 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 Runge-Kutta schemes, wave equations, interface problems, level set method, Hamilton-Jacobi equations, discontinuous Galerkin methods, Stokes problem, Navier-Stokes 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 | Non-convex flux functions
|
| 02/15/2022 Lec 9 | Nonlinear stability theory
|
| 02/17/2022 Lec 10 | High-order 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 | Hamilton-Jacobi equations, numerical methods
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| 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 | Navier-Stokes equations
Read:
Navier-Stokes equations
|
| 04/14/2022 Lec 26 | Projection methods, staggered grid methods
Read:
Projection method
|
| 04/19/2022 Lec 27 | Semispectral methods for the Navier-Stokes 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 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. |
| 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 one-dimensional 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 Shu-Osher 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 Navier-Stokes equations in a 2d box Solves the incompressible Navier-Stokes 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 Navier-Stokes in a periodic 2d box Solves the incompressible Navier-Stokes equations in a rectangular domain with periodic boundary conditions, using a semispectral method and the fast Fourier transform. (Code 2008 by Jean-Christophe Nave) |
| Additional Course Materials | |
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| Homework Problem Sets | |
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| Course Projects | |
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Project proposals due: January 20, 2022.
Project midterm reports due: March 10, 2022. Project final reports due: May 3, 2022.
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