Doc update

README.md

@@ -8,25 +8,54 @@
 4. Test Cases
 
 # 1. Introduction
-DSMC implementation using an octree as a variable mesh.
+DSMC implementation using an octree as a variable mesh based on https://doi.org/10.1016/j.jcp.2008.04.038.
 Implementation in done in python 3.10.
 
 # 2. Requirements
 - python 3.10.
-- pip
+- pip3
+- numpy
+- llvmlite
+- scipy
+- numba
 
 # 3. Installation
-ToDo
+The module can be installed using pip3 from the repository root as
+
+``
+pip3 install .
+``
 
 # 4. Test Cases
+All test cases were performed using Argon.
+The gas properties are as follows:
+
+| gas | $\sigma_T / m^2$ | $m / kg$   | 
+|:---:|:----------------:|:----------:|
+| Ar  | 3.631681e-19     | 6.6422e-26 |
 
 ## 4.1 Heat Bath
-ToDo
+Simulation of temperature relaxation of Argon in closed domain.
+The simulation domain is cube with a side length of $2 \cdot 10^{-3} m$.
+The simulation properties are as follows
+
+
+| $\Delta t / s$ | $w$    | $T / K$ | $n / m^{-3}$ | $u / (m/s)$ |
+|:--------------:|:------:|:-------:|:------------:|:-----------:|
+| 1e-5           | 0.5e-8 | 300     | 1e+20        | 1000.0      |
+
+where $\Delta t$ is the time step, $w$ is the particle weight, $T$  the temperature, $n$ the number density and $u$ the velocity in x direction.
+The simulation results can be seen below.
 
 ![Heat Bath](./examples/heat_bath/heat_bath.png)
 
 ## 4.2 Hypersonic flow around cube
-ToDo
+Hypersonic flow around a cuboid.
+The parameters are as follows
+
+| $\Delta t / s$ | $w$      | $T / K$ | $n / m^{-3}$ | $v_{x, z} / (m s^{-1})$ | $v_y / (m s^{-1})$ |
+|:--------------:|:--------:|:-------:|:------------:|:-----------------------:|:------------------:|
+| 1e-6           | 0.25e+15 | 273.0   | 2.6e+19      | 0                       | -3043.0            |
 
 ![hypersonic_flow](./examples/hypersonic_flow/hypersonic_flow.png)
 
@@ -36,5 +65,44 @@ ToDo
 
 ## 4.3 Shock Tube
 
-ToDo
-![shock Tube](./examples/shock_tube/shock_tube.png)
+This test case is Sod's shock tube problem.
+Initial conditions for the left hand side $C_L$ and the right hand side $C_R$ are found below
+
+$$
+C_L = 
+\begin{pmatrix}
+n_L \\
+u_L \\
+T_L \\
+\end{pmatrix} =
+\begin{pmatrix}
+2.41432e22 \\
+0 \\
+300 \\
+\end{pmatrix}
+$$
+
+$$
+C_R = 
+\begin{pmatrix}
+n_R \\
+u_R \\
+p_L \\
+\end{pmatrix}=
+\begin{pmatrix}
+2.41432e21 \\
+0 \\
+300 \\
+\end{pmatrix}
+$$
+
+The simulation parameters
+
+| $\Delta t / s$ | $w$  |
+|:--------------:|:----:|
+| 1e-7           | 1e-8 |
+
+The simulation domain is a rectangular tube with a square cross section with the side length $2 \cdot 10^{-4} m$ and a length of $0.1 m$.
+Results can be seen below.
+
+![shock Tube](./examples/shock_tube/shock_tube.png)

examples/heat_bath/heat_bath.py

@@ -70,7 +70,7 @@ def calc_x(velocities, mass):
     ax1.set_xlim([0, xmax])
     ax1.set_ylim([0, 0.00040])
 
-    fig.suptitle("T = {:.3f}K".format(Tnew))
+    fig.suptitle("Argon Heat Bath")
     plt.show() 
 
     print('done')