After the course the student is able to
- Create a numerical method for a partial differential equation (PDE) based on a finite difference or finite element discretization.
- Indicate the character of a second order PDE (elliptic, parabolic or hyperbolic) and determine what type of discretization method to use.
- Identify whether or not a special treatment for advection terms is needed.
- Perform a numerical time integration for an evolution problem.
- Carry out a Fourier Stability Analysis for parabolic and hyperbolic problems.
After a short recapitulation of the required mathematics, the character of linear second order PDE’s is determined via the eigenvalues of the corresponding (symmetric) coefficient matrix. It is shown that any second order PDE can be transformed into the Laplace equation, the heat equation or the wave equation, which are representative for elliptic, parabolic and hyperbolic equations, respectively.|
Thereafter, discretization methods are introduced. Both the finite difference method and the finite element method for the discretization of the spatial derivatives are discussed in depth. First the elliptic problems are treated followed by treatment of advection terms.
For the time discretization several explicit and implicit methods are given. It is shown that for parabolic and hyperbolic problems the spatial part can be discretized in a decoupled manner from the temporal part, where finite difference or finite element methods can be used for the former and the time integration schemes for the latter.
The consistency, convergence and stability of numerical methods will be evaluated via Fourier Stability Analysis.
This is a mathematical oriented course. Working knowledge of Math B1, B2, C1, D1 and D2 is assumed.