# PDEs - Mathematical Modeling and Numerical Simulation

Professor: Rolf Krause
Assistant: Lea Conen

Tipo di corso: Lecture
Valore in crediti ECTS: 6
Riferimenti bibliografici sul sito della biblioteca (CoRe)

Many phenomena occurring in real life applications (i.e. physics, finance, biology…) are modeled by means of ordinary (ODE) and partial (PDE) differential equations. These mathematical models are usually sets of equations and relations, which describe the essential behavior of a natural or artificial system, in order to forecast and control its evolution [1].

Objectives
The aim of the course is twofold. Firstly, we will give the students an overview on the construction of differential mathematical models for some basic physical applications; these examples of models will turn out to be particular cases of the more generic class of so called conservation laws. Then, focusing on the arising PDEs, their theoretical mathematical background will be discussed. As a matter of fact, the understanding of PDEs is closely connected to understand their physical meaning and the qualitative and quantitative behaviour of their solutions. This, whenever dealing with a certain PDE, we will directly introduce a numerical solution method, which will allow for studying the behaviour of the PDE under consideration numerically. In this way, theoretical and practical aspects of deriving and solving PDEs will be carefully intertwined. For example, we will deal with the weak formulation of the PDEs, which form the mathematical framework of state of the art simulation methods – such as the finite element method – employed for numerical solution. Theoretical findings will thus be accompanied by doing numerical experiments and by learning about modern solution algorithms. Particular attention will be paid to the applied aspects and implementation activities: to this purpose, whenever possible, the mathematical subjects will be presented in a more practical and intuitive manner (rather than a purely formal one).

Contents

• Fourier law of heat conduction
• Introduction to Fluid Dynamics
• Introduction to Finite Difference method
• Hilbert spaces
• Weak formulation of elliptic problems
• Introduction to Finite Element method
• Weak formulation of evolution problems
• Heat Equation
• Numerical Simulation of the heat equation
• Advection and reaction dominated problems

Teaching mode
The course will be based on lectures and exercise sessions, in which students are asked to participate actively. The students will also implement the discussed methods and will exploit their behaviour in numerical experiments. Moreover, on the basis of numerical experiments, a quantitative understanding of the treated PDEs will be achieved.
Other important and required activities are private study and readings.

References
S. Salsa, Partial Differential Equations in action, from Modeling to Theory, Springer, 2008
A. Quarteroni, Numerical models for differential problems, 4th edition, Springer.
S. Larsson and V. Thomée, Partial Differential Equations with Numerical Methods,
Texts in Applied Mathematics 45, Springer, 2003. Corrected second printing 2005