Numerical Solution of Differential Equations

Objectives:

The course deals with the numerical solution of differential equations and systems of non-linear equations.

Content:

Multistep methods as well as Runge-Kutta method for timedependent problem will be examined. Covergence, order and stability properties will be analysed. For boundary value problems we will have a look at finite difference, finite element and spectral methods.

For solving system of non-linear equations we will study fixpoint iteration and Newton's method, and discusse their convergence properties.

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• has knowledge of state-of-the-art numerical methods in the field.
• Knows the convergence conditions for the different methods.
• Knows which order the different methods have and what exactly the term order means.
• Understands the concept of stiff differential equations, A-stability and stability domain for the different numerical schemes.
• Knows different spatial discretization schemes, such as: finite differences, finite elements and spectral methods.

Skills

The student

• is able to used the methods in numerical calculations. That is; to be able to implement them on a computer.
• is able to analyse the order of a numerical method.
• Understands the possibilities and the limitations of the different methods.

General competence

The student

• is able to make intelligent choices of method for specific problems
• converses easily and unforced about topics such as pros and cons in explicit vs implicit methods