# Efficient Spectral Deferred Correction Methods for Second Order ODEs

Betreuer/in:            Sebastian Götschel
Dekanat/Institut:   Elektrotechnik, Informatik und Mathematik / Mathematik

E-Mail:   sebastian.goetschel@tuhh.de

Motivation:
Time-dependent differential equations are an important modeling tool in computational science and engineering. They are typically solved using numerical algorithms like Euler’s method or Runge-Kutta methods. If high precision solutions are required, the use of methods with high convergence order can be beneficial. While Runge-Kutta methods require order conditions which often are difficult to obtain, Spectral Deferred Correction (SDC) methods are iterative methods that can achieve arbitrary order by changing runtime parameters. They are, however, computationally expensive.

To obtain more efficient SDC methods for stiff problems, modified iterations were introduced in [1] for reaction-diffusion systems, and extended to convection-diffusion equations in [2]. The extension of these techniques to second order equations like Newton’s equations of motion is an open question. Such equations can be used to simulate, e.g., the behavior of a charged particle in a magnetic field, or the movement of planets in the solar system.

Methodolody:
Using Python or Matlab, the project will start by developing a prototype SDC implementation. The approach for achieving more efficient iterations is based on linear algebraic considerations and numerical optimization, to derive suitable approximate iteration matrices. After obtaining a thorough understanding of these techniques, they will be transferred to second order equations. The use of machine learning techniques will be investigated to choose suitable iteration matrices, e.g., depending on solution characteristics.

Aims and deliverables:
* develop good understanding of SDC methods and approaches to increase efficiency
* transfer and extend these approaches to second order equations
* implement these techniques using Python or Matlab
* investigate their performance using suitable numerical experiments

[1] Martin Weiser: Faster SDC convergence on non-equidistant grids by DIRK sweeps. In: BIT Numer. Math. 55 (4) pp. 1219–1241, 2015.
[2] Felix Binkowski: On the convergence behavior of spectral deferred correction methods for convection-diffusion equations. Master’s thesis TU Berlin, appeared as ZIB-Report ZR-17-53, 2017. http://nbn-resolving.de/urn:nbn:de:0297-zib-66016