Remove initial conditions in README's from previous commit

Author: Mark Zhang

examples/2D_kelvin_helmholtz/README.md

@@ -1,36 +1,8 @@
 # Kelvin-Helmholtz Instability (2D)
 
-Reference:
+Reference: See Example 4.8.
 > A.S. Chamarthi, S.H. Frankel, A. Chintagunta, Implicit gradients based novel finite volume scheme for compressible single and multi-component flows, arXiv preprint arXiv:2106.01738 (2021).
 
-**Key Parameters**
-
-* $\gamma = 5/3$
-
-**Initial Conditions**
-
-* **Spatial Domain:** $(x, y) \in [0, 1] \times [0, 1]$
-
-$$
-p = 2.5, \rho(x, y) = \begin{cases}
-2, \text{if } 0.25 < y \leq 0.75\\
-1, \text{else, }\\
-\end{cases}
-$$
-
-$$
-u(x, y) = \begin{cases}
-0.5, \text{if } 0.25 < y \leq 0.75\\
--0.5, \text{else, }
-\end{cases}
-$$
-
-$$
-v(x, y) = 0.1 \sin(4\pi x) \left[ \exp\left( -\frac{(y-0.75)^2}{2\sigma^2} \right) + \exp\left( -\frac{(y-0.25)^2}{2\sigma^2} \right) \right]
-$$
-
-where $\sigma = \frac{0.05}{\sqrt{2}}$.
-
 ### Initial State 
 <img src="figure0.png" height="MAX_HEIGHT"/>
 

examples/2D_richtmyer_meshkov/README.md

@@ -1,41 +1,7 @@
 # Richtmyer-Meshkov Instability (2D)
 
-Reference:
-> A.S. Chamarthi, S.H. Frankel, A. Chintagunta, Implicit gradients based novel finite volume scheme for compressible single and multi-component flows, arXiv preprint arXiv:2106.01738 (2021)., see Example 4.18
-
-**Key Parameters**
-
-* $\lambda = 1.0$
-* $Re = 1e4$
-
-**Initial Conditions**
-
-* **Spatial Domain:** $(x, y) \in [0, 16\lambda] \times [0, \lambda]$
-
-The domain is divided into three regions: Post-shock Air, Pre-shock Air, and SF6. The shock position is given by $x_{shock}=0.7\lambda$. The interface position $x_{int}(y)$ is defined by the sinusoidal perturbation:
-
-$$
-x_{int}(y) = \lambda \left[ 0.4 - 0.1 \sin\left( 2\pi \left( \frac{y}{\lambda} + 0.25 \right) \right) \right]
-$$
-
-The initial state is given by:
-
-$$
-(\rho, u, v, p, \gamma) =
-\begin{cases}
-   (1.4112, 0.8787, 0, 1.6272/1.4, 1.4) & \text{if } x > x_{shock} \quad \text{(Post-shock Air)} \\
-   (1.0, 1.24, 0, 1/1.4, 1.4) & \text{if } x_{int}(y) < x \leq x_{shock} \quad \text{(Pre-shock Air)} \\
-   (5.04, 1.24, 0, 1/1.4, 1.093) & \text{if } x \leq x_{int}(y) \quad \text{(SF6)} \\
-\end{cases}
-$$
-
-The initial perturbation is then smoothed by 
-$$f_{sm} = \frac{1}{2}\left(1 + \text{erf}\left(\frac{\Delta D}{E_i\sqrt{\Delta x \Delta y}}\right)\right)$$
-
-$$u = u_L(1 - f_{sm}) + u_Rf_{sm}$$
-
-where $u$ is each primitive variable. $E_i$ is a thickness constant, and $\Delta D$ is the signed horizontal distance from $x_{int}(y)$. $u_L$ and $u_R$ are left and right interface conditions.
-
+Reference: See Example 4.18.
+> A.S. Chamarthi, S.H. Frankel, A. Chintagunta, Implicit gradients based novel finite volume scheme for compressible single and multi-component flows, arXiv preprint arXiv:2106.01738 (2021).
 
 ### Initial State
 <img src="figure0.png" height="MAX_HEIGHT"/>

examples/2D_viscous_shock_tube/README.md

@@ -1,25 +1,8 @@
 # Viscous Shock Tube (2D)
 
-Reference: 
+Reference: See Example 4.13.
 > A.S. Chamarthi, S.H. Frankel, A. Chintagunta, Implicit gradients based novel finite volume scheme for compressible single and multi-component flows, arXiv preprint arXiv:2106.01738 (2021)., see Example 4.13
 
-**Key Parameters**
-
-* $\gamma = 7/5$
-* $Re = 1000$
-
-**Initial Conditions**
-
-* **Spatial Domain:** $(x, y) \in [0, 1] \times [0, 0.5]$
-
-$$
-(\rho, u, v, p) =
-\begin{cases}
-   (120, 0, 0, \frac{120}{\gamma}) & \text{if } 0 < x < 0.5 \\
-   (1.2, 0, 0, \frac{1.2}{\gamma}) & \text{if } 0.5 \leq x < 1
-\end{cases}
-$$
-
 ### Initial State 
 <img src="figure0.png" height="MAX_HEIGHT"/>