examples: Improve ADER explanation

examples/seismic/tutorials/16_ader_fd.ipynb

@@ -23,7 +23,21 @@
     "\n",
     "$\\frac{\\partial \\mathbf{U}}{\\partial t} = \\begin{bmatrix}\\rho c^2 \\boldsymbol{\\nabla}\\cdot\\mathbf{v} \\\\ \\frac{1}{\\rho}\\boldsymbol{\\nabla}p \\end{bmatrix}$,\n",
     "\n",
-    "expressions for higher-order time derivatives of the state vector are derived in terms of spatial derivatives. High-order explicit timestepping is achieved by substituting these expressions into the Taylor expansion, truncated at the desired temporal discretisation order. As such, the order of the temporal discretisation can be increased to that of the spatial discretisation."
+    "expressions for higher-order time derivatives of the state vector are derived in terms of spatial derivatives. For example, taking the second derivative of the state vector with respect to time\n",
+    "\n",
+    "$\\frac{\\partial^2 \\mathbf{U}}{\\partial t^2} = \\begin{bmatrix}\\rho c^2 \\boldsymbol{\\nabla}\\cdot\\frac{\\partial \\mathbf{v}}{\\partial t} \\\\ \\frac{1}{\\rho}\\boldsymbol{\\nabla}\\frac{\\partial p}{\\partial t} \\end{bmatrix}$.\n",
+    "\n",
+    "Substituting the temporal derivatives on the right hand side for the expressions given in the governing equations\n",
+    "\n",
+    "$\\frac{\\partial^2 \\mathbf{U}}{\\partial t^2} = \\begin{bmatrix}\\rho c^2 \\boldsymbol{\\nabla}\\cdot\\left(\\frac{1}{\\rho}\\boldsymbol{\\nabla}p\\right) \\\\ \\frac{1}{\\rho}\\boldsymbol{\\nabla}\\left(\\rho c^2 \\boldsymbol{\\nabla}\\cdot\\mathbf{v}\\right) \\end{bmatrix}$.\n",
+    "\n",
+    "Assuming constant $c$ and $\\rho$, this simplifies to\n",
+    "\n",
+    "$\\frac{\\partial^2 \\mathbf{U}}{\\partial t^2} = \\begin{bmatrix}c^2 \\nabla^2 p \\\\ c^2\\boldsymbol{\\nabla}\\left(\\boldsymbol{\\nabla}\\cdot\\mathbf{v}\\right) \\end{bmatrix}$.\n",
+    "\n",
+    "This process is iterated to obtain the required temporal derivatives.\n",
+    "\n",
+    "High-order explicit timestepping is achieved by substituting these expressions into the Taylor expansion, truncated at the desired temporal discretisation order. As such, the order of the temporal discretisation can be increased to that of the spatial discretisation."
    ]
   },
   {