+---
+title: "Swift-Hohenberg Solver Write-up"
+bibliography: references.bib
+link-citations: true
+---
+
# Introduction
This program is used to solve the generalized Swift-Hohenberg equation
reasonably well behaved for small values of $r$ and $g_1$, but there
are interesting behaviors that occur when $g_1$ is smaller or larger
than $r$ in magnitude, so this allows us room to vary $g_1$ and
-explore these behavior. To summarize, this code solves:
+explore these behavior. Additionally, we choose $r = 0.3$ because this matches the parameters used by Gurevich in \cite{Gurevich}. We chose our parameters to match so that we could compare the output of our program to the results presented in \cite{Gurevich}, which was useful for validating that our code was functioning properly during the developement process. To summarize, this code solves:
@f{align*}{
\frac{\partial u}{\partial t} = 0.3u - (1 + \Delta)^2 u + g_1 u^2 - u^3
![image](./doc/images/Sphere_Random_Table.png)
-![image](./doc/images/Sinusoid_Random_Table.png)
\ No newline at end of file
+![image](./doc/images/Sinusoid_Random_Table.png)