In this lecture, we construct and analyze efficient numerical integration schemes for highly oscillatory Klein-Gordon type and related equations. The understanding of the theoretical basis of these methods from the lecture shall be deepened in the accompanying problem class by theoretical exercises and as well practical implementation of simple examples with the software MATLAB (or Python).

Please register here for the participation in the problem class. Additional material to the lecture/problem class will be provided there.

As a student of KIT you can get a free license key for *MATLAB* at the KIT software shop. Then you can download the software from mathworks.com after registration.

Python is a free software with a syntax similar to the one of MATLAB.

# Exercises

Please register here for the participation in the problem class. Additional material to the lecture/problem class will be provided there.

Exercise Sheet 1 Program Files uploaded to the internal area

Exercise Sheet 2 Program Files uploaded to the internal area

Exercise Sheet 3 Program Files (and auxiliary material) uploaded to the internal area

Exercise Sheet 4 Auxiliary material uploaded to the internal area

Exercise Sheet 5

# Motivation

The nature of highly oscillatory oscillatory problems can be explained with a simple example, the harmonic oscillator. Initially, a mass attached to a spring with stiffness is deflected units away from its equilibrium. The moment we release the mass, the system starts to oscillate. The rapidness of the oscillations is thereby strictly determined by the value of , i.e. the larger the faster are the oscillations. The position of the mass (with respect to the equilibrium) at time is then determined via the solution of the second order ODE (here with an initial deflection of 1 unit in downward direction and with initial velocity 0)

Harmonic oscillator with

- a nonstiff spring with (green, left)
- an intermediate spring with (yellow, middle) and
- a stiff spring with (red, right).

A simulation movie of the oscillations can be downloaded here.

# Content of the Lecture

The Klein-Gordon equation is a fundamental equation in physics describing the motion of a spin-less particle and is the model problem in our lecture. Its efficient and robust numerical time integration in the nonrelativistic limit regime, i.e. when the speed of light formally tends to infinity, has been subject to current research in the recent years. In this regime, the highly oscillatory nature of the solution is numerically very delicate since standard time integration schemes require severe restrictions on the time step depending on the small parameter which leads to high computational costs.

In order to overcome this numerical challenge, we construct and analyze two types of numerical integrators to efficiently integrate the Klein-Gordon equation in the highly oscillatory

nonrelativistic limit regime as well as in slowly oscillatory relativistic regime:

- The first type of method relies on the asymptotic behaviour of the solution as which allows us to break down the numerical task to solving a non-oscillatory Schrödinger limit system with a standard Strang splitting scheme. We thus obtain a scheme which satifies error bounds of order without any time step restriction. Despite that this scheme provides brilliant results at very low cost in the highly oscillatory regime , the approximation to the exact solution in the intermediate and slowly oscillatory relativist regimes is limited through the error term of order . This motivates us to construct a second type of integrator which allows uniformly accurate in error bounds.

- The second type of method is based on the idea of exponential integrators applied to so-called "twisted variables". These "twisted variables" satisfy a first order in time differential equation with a bounded (with respect to ) right hand side. Thereby, starting from Duhamel's formula (variation of constants formula) for the latter equation, we integrate the linear terms exactly and approximate the "nice" nonlinearity via a truncated Taylor series expansion. The resulting scheme allows first order in time error bounds in independent of the parameter . We thus call this scheme uniformly accurate in , first order in time.