Adding notes about stability in the high-precision DG1 notebook.

assignment1/dg1_high_prec.ipynb

@@ -22,7 +22,7 @@
     "total_time = mpmath.mpf('1.0')\n",
     "p_order = 3\n",
     "num_intervals = 16\n",
-    "# CFL condition breaks down between 107 and 106 time steps.\n",
+    "\n",
     "dt = mpmath.mpf('1') / 108\n",
     "solver = dg1.DG1Solver(num_intervals=num_intervals,\n",
     "                       p_order=p_order,\n",
@@ -31,6 +31,27 @@
     "                       points_on_ref_int=dg1_high_prec.gauss_lobatto_points)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The CFL / RK condition breaks down between $106$ and $107$ time steps.\n",
+    "\n",
+    "We actually don't expect stability until $N_T = 111$ (number of timesteps).\n",
+    "The stability function\n",
+    "$$R(z) = 1 + z + \\frac{z^2}{2} + \\frac{z^3}{6} + \\frac{z^4}{24}$$\n",
+    "is evaluated at \n",
+    "$$z = \\frac{\\lambda \\Delta t}{\\Delta x} = \\frac{16 \\lambda}{N_T}$$\n",
+    "(since we have $16$ space steps and $N_T$ time steps and both the\n",
+    "space and time intervals are length 1).\n",
+    "\n",
+    "When $p = 3$, the largest magnitude value in the spectrum of the space\n",
+    "discretization is $\\lambda \\approx \\frac{-19.15688121515517}{\\Delta x}$ \n",
+    "and we enter the region of absolute stability when\n",
+    "$$R\\left(\\frac{16 \\lambda}{111}\\right) \\approx 0.96450003389941052 < 1 < \n",
+    "1.0017531696935333 \\approx R\\left(\\frac{16 \\lambda}{110}\\right).$$"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 2,