Clean up boundary conditions and predictor corrector for staggered grid

notebooks/lab7/01-lab7.ipynb

@@ -634,20 +634,25 @@
     "$$\\frac {h_{i}(t+dt)-h_{i}(t-dt)}{2 dt} + H \\frac {u_{i}(t) - u_{i - 1}(t)}{dx} = 0$$\n",
     "</div>\n",
     "\n",
-    "The initial conditions are: At $t = 0$ and $t = dt$, all points have\n",
+    "The initial conditions are: At $t = 0$, all points have\n",
     "zero elevation except at $h_{3}$, where \n",
     "$$h_{3}(0) = h_{0}$$\n",
+    "At $t = dt$, \n",
+    "$$h_{2}(dt) = h_{0} Hg \\frac{dt^2}{2 dx^2}$$ \n",
     "$$h_{3}(dt) = h_{3}(0) - h_{0} Hg \\frac{dt^2}{dx^2}$$ \n",
+    "$$h_{4}(dt) = h_{0} Hg \\frac{dt^2}{2 dx^2}$$ \n",
+    "\n",
     "At $t = 0$ and\n",
     "$t = dt$, all points have zero velocity except at $u_{2}$ and $u_{3}$,\n",
     "where \n",
     "$$u_{2}(dt) = - h_{0} g \\frac{dt}{dx}$$\n",
     "$$u_{3}(dt) = - u_{2}(dt)$$ \n",
     "This time we assume there is a wall at\n",
-    "$u_{1}$ and $u_{4}$, so we will ignore the value of $h_{1}$. The\n",
-    "boundary conditions are: \n",
+    "$u_{1}$ and $u_{4}$. The\n",
+    "boundary conditions in $u$ are: \n",
     "$$u_{1}(t) = 0$$ \n",
     "$$u_{4}(t) = 0$$\n",
+    "We can use $\\partial h/\\partial x=0$ at the wall to put boundary conditions on $h$.  An exercise for the reader is to decide which two $h$ points straddle the walls.\n",
     "\n",
     "### Problem One\n",
     "[lab7:prob:staggered]:(#Problem-One)\n",