Update README.md

README.md

@@ -56,12 +56,14 @@ where \\(\Delta = \nabla \cdot \nabla\\) is the spatial Laplacian and \\(U\\) th
 $\\( U_n(t, \mathbf{x}) = \mathbf{n} \cdot \nabla U = 0, \quad t \in \mathbb{R}, \quad \mathbf{x} \in \partial \Omega. \\)$
 
 Upon taking the temporal Fourier transform, we get the inhomogeneous Helmholtz Neumann boundary value problem
-```math
+
+$$
 \begin{align}
 -(\Delta + \omega^2)u &= \delta_{\mathbf{x}_0}, \quad \text{in } \Omega,\\
 u_n &= 0 \quad \text{on } \partial \Omega,
 \end{align}
-```
+$$
+
 with outwards radiation conditions as described in [1]. The region \\(\Omega\\) lies above a corrugated boundary \\(\partial \Omega\\), extending with spatial period \\(d\\) in the \\(x_1\\) direction, and is unbounded in the positive \\(x_2\\) direction. The current example is a right-angled staircase whose unit cell consists of two equal-length line segments at \\(\pi/2\\) angle to each other.
 
 ![Gif](https://users.flatironinstitute.org/~polymathic/data/the_well/datasets/helmholtz_staircase/gif/pressure_normalized.gif)
@@ -96,12 +98,11 @@ the boundary) are enforced at the boundary.
 **Simulation time-step:** continuous in time (time-dependence is
 analytic).
 
-**Data are stored separated by (\\(\Delta t\\)):** \\(\Delta t =\frac{2\pi}{\omega N}\\), with \\(N = 50\\).
+**Data are stored separated by ( \\(\Delta t\\)):** \\(\Delta t =\frac{2\pi}{\omega N}\\), with \\(N = 50\\).
 
-**Total time range (\\(t_{min}\\) to \\(t_{max}\\)):** \\(t_{\mathrm{min}} = 0\\), $t_{\mathrm{max}} =
-\frac{2\pi}{\omega}$.
+**Total time range ( \\(t_{min}\\) to \\(t_{max}\\)):** \\(t_{\mathrm{min}} = 0\\), \\(t_{\mathrm{max}} = \frac{2\pi}{\omega}\\).
 
-**Spatial domain size (\\(L_x\\), \\(L_y\\), \\(L_z\\)):** \\(-8.0 \leq x_1 \leq 8.0\\) horizontally, and \\(-0.5 \geq x_2 \geq 3.5\\) vertically.
+**Spatial domain size ( \\(L_x\\), \\(L_y\\), \\(L_z\\)):** \\(-8.0 \leq x_1 \leq 8.0\\) horizontally, and \\(-0.5 \geq x_2 \geq 3.5\\) vertically.
 
 **Set of coefficients or non-dimensional parameters evaluated:** \\(\omega\\)={0.06283032, 0.25123038, 0.43929689, 0.62675846, 0.81330465, 0.99856671, 1.18207893, 1.36324313, 1.5412579, 1.71501267, 1.88295798, 2.04282969, 2.19133479, 2.32367294, 2.4331094,  2.5110908}, with the sources coordinates being all combinations of \\(x\\)={-0.4, -0.3, -0.2, -0.1, 0, 0.1, 0.2, 0.3, 0.4} and \\(y\\)={-0.2, -0.1, 0, 0.1, 0.2, 0.3, 0.4}.