Discontinuous Galerkin Study Project

This project is a personal foray into the discontinuous Galerkin method.

Navigating the Repository

• domains stores the code for elements used. Currently only spec_elem.py exists, which sets up single SEM elements in 2D, or a continuous grid in 2D, where multiple elements share edges and basis functions at each node.
• outputs is a database of outputs for run simulations. Unless I make a mistake, significant data should not be placed here -- only things that are worth sharing, like animations.
• problems stores the code for an equation to be solved on the above domains. When applied, these overwrite functions in domains to be able to run a problem.
• report holds notes of my thoughts on the progress I've made. This just serves as a journal/notebook to put my thoughts to paper.
• sims stores the code that can be run to perform a simulation.
• tests are unit tests. Currently, only tests to verify the SEM elements in domains/spec_elem.py exist.

Current Step

The order 1 wave equation was built to try the upwind flux scheme in Igel (2017), but for 2d. The details of the derivation can be found here. The present simulation output, in outputs/order1wave_expand.mp4 took a Gaussian initial condition $$u_0(x,y)= a_0 e^{-\frac{(x - c_x)^2 + (y-c_y)^2}{2\sigma^2}}$$ in two places,

• $\Omega = [-20,20]\times[-10,10]$, with a discontinuous boundary at $x=0$.
• $N = 3$ SEM elements, each of size $1\times 1$ unit.
• $\sigma^2 = 2.5$, with wave centers at $(20/3,5)$ and $(-20/3, 5)$.
• wave speed $c = 2$
• Time is run using Heun's method to $T=10$ using a time step of $\Delta t = 0.01$. Each frame of the animation is a step of $0.1$ time units.