Rational Approximations

Dear Participants,

we have seen in the previous lecture that it is sometimes of advantage to combine operator splitting procedures with other time discretisation methods. All the examples we considered until now, especially the simplest ones, the implicit Euler method and the Crank-Nicolson scheme, are special cases of rational approximations. This is the main topic of the remaining lectures of this course.

The main idea is to approximate the exponential function near zero with nice rational functions and then try to insert the generator into this approximation. As a technical tool, we have to develop a so-called functional calculus which will enable us to do this. We start with the scalar case to motivate the ongoing procedures.

Unfortunately, the less we require from the generator, the more sophisticated methods we have to use to develop this functional calculus. We start this week with multiplication operators and extend the theory next week to sectorial ones. You can read more in the lectures following the pdf link below.

We kindly ask the team of Delft to provide this week's official solutions.

Have fun and good reading,
your virtual lecturers

Alexander
András
Bálint
Petra

ISEM15_Lecture12.pdf

Discussion of Lecture 12.

Solutions and discussion of the exercises.