Numerical Analysis class at MIT taught my David Sanders. Created many differential solvers using Julia
My project proposal: My current work in physics at MIT includes evolving nonlinear differential equation that describe the final steps of Inflation, the seconds following the Big Bang where the universe experienced a period of rapid expansion and cooling. In order to simulate this on a computer we are utilizing DOPRI5, also known as the Dormand Prince method which is a variation of the Runge Kutta method – however my understanding of this method is minimal.
For my final project I would like to explore the use of Linear and Non-Linear Differential Equation solver, to display its many applications and possible errors that arise when doing computational physics. I plan to start my project with rudimentary examples of where RK, showing plots of the motion of pendulums and masses on springs and general simple harmonic motion. I will then build to examples where we add damping and other variables to the ODE. I still must research this further, but I wish to try and see if there is a way to animate plots to show time evolution of the solutions to the ODE. I would also like to eventually find a way to solve coupled systems and have interesting and convoluted solutions, perhaps demonstrating chaos. This is when very small changes in parameters creates bizarre physics. For instance, this could be seen systems that have a pendulum on a spring, or multiples masses on a spring, etc.
The last application that I wish to should is applying RK to solve PDEs. In this regime I will try to evolve some basic PDEs such as the wave equation and perhaps the diffusion equation. I am excited to see how I can make interactive plots where perhaps users can perhaps generate a wave and Pluto can display the evolution of the wave with set boundary conditions. This will require further exploration of packages and perhaps utilizing the adaptive-step size method of RK.
Throughout this project, I wish to compare numerical solutions to the analytical solutions where possible. I can also use the Julia built-ins for solving ODEs to do so. In order to help me, I plan to just books such as Numerical Recipes and Julia documentation for help with packages. I may have to consult some old physics notes to generate meaningful equations which I can then time evolve.