PDE Project Course 04/05

Suggestions for projects

Karin Kraft and Johan Jansson Dept. Mathematical Sciences Chalmers University of Technology

PDE Project Course 04/05  Projects

General guidelines

This document contains a list of projects. Since these are only suggestions, you are welcome with your own ideas. Regard the list as an inspiration, and perhaps a hint on the expected level of your projects.

Concerning grades, the projects are divided into two parts: basic level and advanced level. Basic level means grade 3 and advanced level means grade 4 or 5. However, advanced level is no guarantee for grade 4 or 5. It is also required that your report and your presentation match the level of your project. It is also possible to receive a higher grade even if you only complete the basic level, if you deliver an excellent report and an excellent presentation.

Good luck! Johan and Karin

1 Convection-Diffusion

Implement your own solver for the convection-diffusion equation in 2 dimensions from scratch in Matlab.

Advanced

Extend your solver to 3D.

References

1. Applied Mathematics: Body and Soul, by Eriksson, Estep, and Johnson, Springer Verlag 2003.

2. Computational Differential Equations, by Eriksson, Estep, Hansbo, and Johnson. Studentlitteratur 1996.

2 Chemical reactions

Simulate the following system of chemical reactions, where the substances A and B react to form C: A + B → C.

Consider a beaker containing a solution of A with given concentration. To this beaker, we add a drop of B every second until finally A has “completely” reacted with B. Try to find a suitable reaction to simulate in a chemistry book. Maybe the reaction you want to simulate is instead given by 2A+3B → 4C, or perhaps 5A + 2B + C → 2C?

Model this as a system of reaction–diffusion equations, where u_{1}(x, t) and u_{2}(x, t) are the two concentrations to be determined.

Implement your 2D solver as a module in DOLFIN.

Advanced

Solve the problem in 3D in Dolfin.

References

1. Applied Mathematics: Body and Soul, by Eriksson, Estep, and Johnson, Springer Verlag 2003.

2. Computational Differential Equations, by Eriksson, Estep, Hansbo, and Johnson. Studentlitteratur 1996.

3. Some suitable book on chemistry.

3 Heat equation

Write a solver for the heat equation in 2D from scratch (in Matlab). Compute error estimates (energy norm and/or L_{2}-norm) and study how the solution changes when the mesh size h is changed.

Advanced

Consider one of the following extensions:

• Do the computations in 3D.

• Compute error estimates and refine the mesh manually where the error is large. Study the convergence of the solution.

References

1. Applied Mathematics: Body and Soul, by Eriksson, Estep, and Johnson, Springer Verlag 2003.

2. Computational Differential Equations, by Eriksson, Estep, Hansbo, and Johnson. Studentlitteratur 1996.

4 The Navier-Stokes equations

Implement a solver for the Navier-Stokes equations in 2D or 3D.

Advanced

Nothing extra is needed for advanced level.

References

1. Applied Mathematics: Body and Soul, by Eriksson, Estep, and Johnson, Springer Verlag 2003.

2. Computational Differential Equations by Eriksson, Estep, Hansbo, and Johnson. Studentlitteratur 1996.

3. Adaptive finite element methods for turbulent flow by Johan Hoffman. Chalmers Finite Element Center Preprint 2002–08, available at

http://www.phi.chalmers.se/preprints/.

5 Elasticity

Implement a solver for linear elasticity in Puffin in 2D. Assume that your materials are isotropic (same stiffness in all directions).

Advanced

Extend the solver to also handle anisotropic materials.

References

1. Beyond the Elements of Finite Elements: General Principles for Solid and Fluid Mechanics Applications by Hansbo. Department of Solid Mechanics, Chalmers University of Technology, 2002

6 The Wave equation

Read the technical report by L.Beilina and implement the method in 2D.

Advanced

Implement the absorbing boundary condition.

References

1. A Hybrid Method for the Wave Equation, by L. Beilina, Technical report, Chalmers Finite Element Center (2001)

2. Applied Mathematics: Body and Soul, by Eriksson, Estep, and Johnson, Springer Verlag 2003.

3. Computational Differential Equations by Eriksson, Estep, Hansbo and Johnson. Studentlitteratur 1996.

7 Bistable equation

Write a solver in Dolfin for the bistable equation in 2D, which is an easy example of a nonlinear PDE.

Advanced

Extend the solver to 3D.

References

1. Body and Soul computer sessions (Reaction-Diffusion):

     http://www.phi.chalmers.se/bodysoul/sessions/.