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Goal: reliable hands-on computing with PDEs

When you encounter a partial differential equation, you will after this course have a clear idea of

• how to view the problem as an assemply of smaller/simplified pieces
• how to formulate a finite difference or finite element method for the problem
• how to implement the numerical method in a program
• how to verify that the numerical computations are correct
• how to estimate the reliability of numerical simulations, and in particular discover non-physical numerical artifacts
• how different sources of errors play together

Teaching method: simplify, understand, generalize

To take advantage of the true power of mathematics in physics, biology, and other applied sciences, one needs to simplify, understand, and generalize. These three concepts also form the teaching basis in this course.

A series of physical model problems are addressed and used in examples:

• exponential decay
• vibrating systems
• diffusion
• wave motion
• viscous flow
• coupled flow, deformation, transport, and/or heat transfer

For each model problem, we start out with the simplest version of the model and go through key building blocks for turning the computer into a virtual laboratory where we can play around with the model. The building blocks covered in the course are

• discretization of the model (finite difference, element, volume; staggered grids, mixed spaces; discontinuous Galerkin methods)
• methods for nonlinear differential equations
• implementation of the discrete model (from scratch in 1D, using tools like FEniCS for 2D/3D)
• programming tools: Python, Cython, Python-F77/C++ (via f2py, Cython, or Instant), FEniCS, nose/pytest, SymPy
• verification of the implementation
• analysis of the discrete model: exact (Fourier) solutions, trunction error
• understanding what can go wrong (numerical artifacts)
• extensions, generalizations, real applications
• scaling, real data with units
• parallel computing
• report writing (LaTeX, HTML/MathJax, Sphinx), reproducibility, Git version control, GitHub project hosting service

Much focus is on all the nuts and bolts of building reliable simulation models. This means a particular focus on implementation and testing, and less focus on mathematical subjects that do not have a direct application of great practical value.

When teaching new methods, we always start out with simple ODEs to explain the very basics of the technicalities. Then we extend the methodology progressively using more advanecd models that build on what we have learned through the simpler ones. In this way, new mathematical or programming techniques are introduced in problems of the simplest possible type (but no simpler!), yet of a relevance that carries directly over to advanced PDE problems.

Through the simplify, understand, and generalize paradigm, one will view a complex scientific application as an assembly of pieces that can be understood separately and then brought together (perhaps with considerable efforts!) to solve the problem in a reliable way. Our focus throughout the course is two-fold: 1) how to do it, and 2) how to assess the quality.

Compulsory exercises and projects

• A number of small mandatory exercises.

1. The first mandatory exercises are from Finite difference methods for vibration ODEs. Problem 1: Use linear/quadratic functions for verification and Exercise 21: Simulate an elastic pendulum. The submission deadline is August 29.
2. The second mandatory exercise is Exercise 13 Compare discretizations of a Neumann condition from the module of Finite difference methods for wave motion. The submission deadline is September 12.
3. The third mandatory exercise consists of Exercise 2 and Exercise 5 of the Stationary variational forms module. The submission deadline is October 10.
4. The fourth mandatory assignment is Exercise 8 of the stationary variational forms module. The submission deadline is October 24. (You need only to write a simple report in LaTeX, no programming needed.)
5. The fifth mandatory assignment is Exercise 6 of the sovling nonlinear ODE and PDE problems module. The submission deadline is October 31.

• Two large compulsory projects.

1. The first compulsory project is A 2D wave equation solver with finite differences, programmed from scratch. Deadline for submission is September 19. Group assessement on September 20.
2. The second compulsory project is A nonlinear finite element application, programmed in the FEniCS system. Deadline for submission is November 7. Group assessment on November 8.

These exercises and projects must be submitted before prescribed deadlines in a github.com repo and assessed in the corresponding assessment lectures (under guidance of the teaching assistant), where groups of four students are automatically formed, each student discusses and provides feedback on the exercises done by the three other students. Each student must do both the compulsory projects, and in addition finish at least 3 (out of 5) smaller mandatory exercises.

Files for the course

All example codes are available from https://github.com/hplgit/num-methods-for-PDEs/tree/master/src. All written course material (textbook chapters and lectures) are available at http://hplgit.github.io/num-methods-for-PDEs/doc/pub/index.html.

You may access these resources directly on the web, or you may have a local copy on your computer:

Terminal> git clone https://github.com/hplgit/num-methods-for-PDEs.git INF5620-resources

This gives you a directory INF5620-resources with all the published course material. The written material is in doc/pub and the computer codes in src. As the material undergoes updates throughout the course, you must frequently get these updates by going to the INF5620-resources project and doing:

Terminal> git pull origin master

Exam

See previous exams for examples on topics for the final exam.