CME306 / CS205B
Mathematical Methods for Fluids, Solids and Interfaces (Spring 2009)

Course Announcements

Date                      Contents
2009-6-13Homework 9 solutions are now available.
2009-6-02Lectures 17-19 are available, and Extra Credit 1, Homework 6 and 7 are graded and available.
2009-5-28Extra Credit 2 has been posted.
2009-5-27Typos fixed in homework 9; solutions to homework 7 and 8 posted.
2009-5-26Minor clarification to Homework 9: density also lives at cell centers.
2009-5-25Homework 9, Letures 15 and 16 are now available.
2009-5-19Homework 5 has been graded and is available at Gates 377.
2009-5-18Homework 8 and Lectures 13 and 14 are now available.
2009-5-14Solutions to Homework 4,5 and 6 are now available.
2009-5-13Homework 7 is now available. Please note that it is due Tuesday.
2009-5-12Lecture 12 and Extra Credit assigment 1 are now available.
2009-5-7Lectures 10 and 11 are now available.
2009-5-7Homework 6 is posted. Note that it is due Thursday next week.
2009-5-5Lecture 10 has been posted.
2009-4-30Homework 3 and 4 have been graded and are now available at Gates 377.
2009-4-28Lecture 9 and Homework 5 are posted.
2009-4-26Homework 4 has a typo; the online version is now correct.
2009-4-25Homework 3 solutions have been posted.
2009-4-21Lectures 6, 7 and 8 have been posted, and homework 4 is now available.
2009-4-15Homework 2 has been graded and is available at Gates 377, and solutions have been posted.
2009-4-14Lecture 5 and Homework 3 are posted. Graded homework 1 will be available by the end of the day, and solutions to homework 2 will be posted by noon.
2009-4-10Lecture 4 is posted.
2009-4-09Homework 1 solutions posted.
2009-4-08Several mistakes in the homework assignment have been fixed. In particular, you are asked do rederive the weak form of conservation of mass, not momentum in the first section of the assignment, and the method is Runge, not Runga.
2009-4-08Discussion section notes will not be posted this week. The material covered was the Lax-Richtmyer framework (which is covered in Discussion 1), and Runge-Kutta (of which everything I covered is provided in the homework itself.
2009-4-07Homework 2 and Lecture 3 are posted.
2009-4-02Several minor changes have been made to the Discussion Section; please read the description and note that an additional section has been added that will cover the same material as the Wednesday section.
2009-4-02Lecture 2 is posted.
2009-4-02The theory section has been updated -- please note that Jón will be covering his material again during Friday office hours.
2009-3-31Homework 1 is posted.
2009-3-31Website is live!


Overview of numerical methods for the simulation of problems involving solid mechanics and fluid dynamics. The focus is on practical tools needed for simulation, as well as the necessary continuous mathematics involving nonlinear hyperbolic partial differential equations. Possible topics include the finite element method, highly deformable elastic bodies, plasticity, fracture, the level set method, Burgers' equation, compressible and incompressible Navier-Stokes equations, smoke, water and solid-fluid coupling.


Please refer all questions about course material and practices to the CAs before contacting Professor Fedkiw. If you have a question for the CAs, please make sure that it isn't answered on this webpage before contacting them. Also, please do not show up outside of scheduled office hours without first making an appointment. When emailing the CAs, make sure to include "CME306" or "CS205B" somewhere in the subject of your message.

Meeting Times

Useful Texts

Class Notes

Lecture 1 Introduction, Simulation of Materials
Lecture 2 Conservation of Mass, Smoothed Particle Hydrodynamics
Lecture 3 Smoothed Particle Hydrodynamics, Forces, Linearized System
Lecture 4 Ordinary Differential Equations, Stability, Newmark Methods
Lecture 5 Springs
Lecture 6 Springs
Lecture 7 Finite Element Method
Lecture 8 Finite Element Method, Rigid Bodies
Lecture 9 Advection, Runge-Kutta, Hamilton-Jacobi ENO, Semi-Lagrangian Advection
Lecture 10 Hyperbolic Conservation Laws, Shocks and Rarefactions
Lecture 11 Discrete Conservation Form, ENO-Roe, ENO-LLF
Lecture 12 Discrete Conservation Form, ENO-Roe, ENO-LLF
Lecture 13 Multiple Dimensions, Systems
Lecture 14 Systems, Discretizations, Shallow Water
Lecture 15 Incompressible Flow Equations
Lecture 16 Incompressible Flow Equations: the Poisson Equation
Lecture 17 Incompressible Flow Equations: Discretization, Semi-Lagrangian Advection
Lecture 18 Heat Equation
Lecture 19 Viscosity, Vorticity

Discussion Section Notes

In addition to lecture, there will be a section run on Wednesday from 5:30PM to 6:30PM and repeated on Friday from 3:00PM to 4:00PM, which covers tools necessary for the completion of the theory-track homework. While not manditory, it is strongly recommended for any students who intend to take the iCME Qualifying Exams.

Discussion 1Convergence, Consistency and Stability, Von Neumann analysis, and the Lax-Richtmyer Equivalence Theorem
Discussion 2Modified Equations, Dissipation vs. Dispersion, Rigid Body Rotations


There will be a problem set assigned each week which will be posted on Tuesday at 11:00AM. There are two separate tracks of homework: theory (which involves more qualifying exam preparation), and application (which involves more software development). If you are taking the iCME qualifying exams, you are strongly encouraged to take the theoretical track. If you are looking for the application homework, take a look at CS 205B.

Homework will be graded in coarse, half-point increments between 0 and 2 points.

You may collaborate on homework assignments provided each student writes up his or her own solutions and clearly lists the names of all the students in the group.

TheoreticalDue Date
Homework 1(tex) (Solution) 2008-4-7
Homework 2(tex) (Solution) 2008-4-14
Homework 3(tex) (Solution) 2008-4-21
Homework 4(tex) (Solution) 2008-4-28
Homework 5(tex) (Solution) 2008-5-5
Homework 6(tex) (Solution) 2008-5-14
Homework 7(tex) (Solution) 2008-5-19
Homework 8(tex) (Solution) 2008-5-26
Homework 9(tex) (Solution) 2008-6-2

Homework must be submitted physically either in class or in the bin outside Gates 204 by 11 AM. Graded homework will be available for pickup in the Gates 377 filing cabinets.

Extra Credit

Depending on class performance on homework and exams, there will be a small number of optional extra credit assignments. These assignments will require you to implement some of the numerical schemes discussed during the course. Extra credit projects do not have a set point value; rather, they will be taken into account at the end of the quarter when determining your final letter grade.

Please note that you are free to do, as extra credit, the homework from the applied track.

AssignmentDue Date
Extra Credit 1(tex)2008-5-26
Extra Credit 2(tex)2008-6-4


The final grade will be calculated based on the highest score between your theoretical and applied homework. The lower of the two will be counted towards extra credit. You should complete either all of the theoretical or all of the applied homework, not a mix of the two.

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