Course syllabus

This graduate course presents the theory and application of numerical approaches to solution of differential equations. Both ordinary differential equations (ODEs) and partial differential equations (PDEs) are discussed.

Course goals: students will acquire proficiency in the formulation of numerical schemes for solving ODEs and PDEs using finite difference, finite volume, finite element, boundary element, and spectral methods. A broad overview of each approach will be discussed. Depending on class interest, specific methods will be studied in more detail. Application are chosen from domains such as fluid dynamics, rheology, elasticity, plasticity.

Honor Code:

• Unless explicitly stated otherwise, all work is individual. You may discuss various approaches to homework problems with students, instructors, but must draft your answers by yourself. In joint projects, each student will clearly identify which portions of the work they contributed.

Grading

Required work

• Homework - Best 10 of 12 assignments x 6 = 60 points

• Midterm examination = 9 points

• Final project = 21 points

• Extra credit - additional two homework assignments, 2 x 6 = 12 points

Mapping of point scores to letter grades

Course policies

• Students are free to establish their own schedule; there is no need to inform instructor of absences. Course attendance is highly recommended to gain insight into course topics

• Late homework is not accepted.

• Homework is to be submitted electronically through Sakai

Examinations

• The midterm examination during normal class meeting time before Fall Break will consist of 3 questions, 3 points each.

• The final examination will consist of 7 questions, 3 points each.

Course materials

Bibliography

Advanced Engineering Mathematics, E. Kreyszig

Class notes

Homework

Course topics

• Ordinary differential equations:

  • Convergence, consistency, stability

  • Single-step, multi-step methods

  • Boundary value problems

• Finite difference methods and analysis:

  • Convergence, consistency, stability

  • Modified equations

  • Fourier analysis of finite difference methods

• Finite volume methods:

  • Godunov methods and Riemann problems

  • Essentially non-oscillatory schemes

  • Central schemes

  • Adaptive mesh refinement

• Spectral methods:

  • Operator eigenfunction expansions

  • Collocation methods

  • Riesz theorem

  • Convergence analysis

• Finite element methods

  • Grid generation

  • Finite element spaces

  • System matrix assembly

  • Variational formulations

SciComp@UNC Linux environment

Scientific computation is typically carried out in a Un*x environment (e.g. OS/X, various Linux versions). This course uses a customized Linux environment named SciComp@UNC available to students as a virtual machine. Download Virtual Box and the SciComp@UNC virtual machine image.

Various open source tools for carrying out and documenting practical scientific computation will be successively introduced:

• TeXmacs

• SciPy

• Maxima

• Octave

• Gnu compilers (Fortran,C,Go)

• Julia

• Paraview

• OpenDX

• Gnuplot

• BEARCLAW

The course will also use a few commercial tools, freely available to students while connected to the campus network (either directly or remotely through the UNC VPN server):

• Mathematica

• Totalview

Course material repository

Course materials (lecture notes, workbooks, homework, examination examples) are stored in a repository that is accessed through the subversion utility, available on all major operating systems. The URL of the material is http://mitran-lab.amath.unc.edu/courses/MATH761

The above address is needed for an initial checkout using commands such as:

In the SciComp@UNC virtual machine the initial checkout can be carried out through the terminal commands

Update the course materials before each lecture by:

Links to course materials will also be posted to this site, but the most up-to-date version is that from the subversion repository, so carry out the svn update procedure prior to each lecture.