Symplectic Integrators
Introduction
Many applications require the use of an integration method that preserves the energy over the course of a simulation. Hence, the aim of this project is to implement multiple methods which do not accumulate errors in energy and phase space volume over a large number of integration steps.A famous class of methods which fulfill the above criteria are the symplectic integrators.
Getting Into The Program
After going through the list of projects, I decided upon the one I was going to apply, pretty early on. I was really interested in the physics behind the symplectic integrator. I also had some prior experience of working on implementing them. I mailed the mentors with my interest in the project, following which I received a qualification task. I submitted my solution to the task and after that, worked with my mentors to develop a project proposal. The projects were announced in may and I was selected!
Starting Off - The Community Bonding Period
This was a very essential period. I had my first meet with my mentors and I got to know a bit about their works. We figured out the logistics of how we’ll be planning out the 3 month long coding period. I also got to attend the GSOC’22 Global Summit and learnt about the importance of open source. The rest of the community bonding period was utilized on brushing up my basics as well as learning to use the Geant4 toolkit.
Coding Period - Kick Off
Before starting work on the actual methods, it was important to develop a testbed application that could be used to test the steppers and drivers. Hence, I started off my coding period by implementing a geant4 application to track a particle in a geometry initialized with a magnetic field. This was really interesting and exciting for me as I had done my homework during the community bonding period. The testbed is comprised of a detector constructor class, a physics list class and a primary action generator class. The magnetic field of the geometry and the stepper used for tracking the particle can be defined in the detector constructor. The physics list is used to define various particles and physical processes. Finally, the primary generator was used to initialize the particle. I utilized the test bed to test out various steppers already present in the codebase to profile them.
First Method
After implementing the above application, we started deliberating on the first method that we’ll implement. We ultimately decided on the Boris Algorithm for its simplicity and wide use. The same was implemented by using the familiar driver and stepper design in Geant4. The stepper(G4BorisScheme) implements the actual Boris Algorithm and the driver(G4BorisDriver) is used to manage the stepper.
Second Method
The Boris method is a second order method. This means that the magnitude of the numerical error is going to vary as the square of the step size of the integration. To improve upon this, we decided to implement a higher order method to reduce the error due to the step size.Boris-SDC is fundamentally a collocation method solved via spectral deferred corrections, using the same trick as the Boris algorithm to avoid an implicit velocity dependence. In such methods, we establish interpolating polynomial that satisfy the fundamental ODE equation and then integrate these polynomial via quadratures.Furthermore, the method uses Spectral Deferred Corrections to make converging approximations of the collocation solution.
Implementing Boris-SDC
In order to understand and implement the method, I studied the thesis by Kristoffer Smedt on High-Order Particle Integration for Particle-In-Cell Schemes.
The method was implemented in a similar style to the original Boris method. The stepper(G4BorisSDC) implements the actual algorithm and the driver(G4BorisDriverSDC) is used to manage the stepper.The method still requires some correction.
Overall Experience
I had an absolutely wonderful experience this summer. This wan a highly educative opportunity and I learnt a lot about open source.It gave me an insight into how large projects are maintained .I would like to express my gratitude to my mentors John Apostolakis, Soon Jung Yun and Renee Fatemi, for they have been extremely supportive and helpful throughout the program.