----
-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. 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:
+explore these behavior. Additionally, we choose $r = 0.3$ because this matches the parameters used by Gurevich in [1]. We chose our parameters to match so that we could compare the output of our program to the results presented in [1], 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)
+
+# References
+
+[1] Svetlana Gurevich. *Chapter 4: Swift-Hohenberg Equation*. url : https://www.uni-muenster.de/Physik.TP/archive/fileadmin/lehre/NumMethoden/SoSe10/Skript/SH.pdf.