work on parabolic module

content/pdes/parabolic.ipynb

@@ -239,27 +239,39 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Implicit scheme\n"
+    "## Implicit scheme\n",
+    "\n"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
+    "clear all\n",
+    "\n",
+    "alpha = 2.3e-1;\n",
     "dx = 0.1;\n",
+    "x = [0 : dx : 1]; n = length(x);\n",
+    "\n",
+    "% choose a Fourier number that is deliberately past the explicit method stability limit\n",
     "Fo = 0.75;\n",
-    "alpha = 2.3e-1;\n",
     "dt = Fo * dx^2 / alpha;\n",
     "\n",
-    "x = [0 : dx : 1]; n = length(x);\n",
-    "t = [0 : dt : 10]; m = length(t);\n",
+    "t = [0 : dt : 1]; m = length(t);\n",
     "\n",
     "T = zeros(m, n);\n",
+    "\n",
+    "% Initial conditions\n",
     "T(1,:) = 200;\n",
     "\n",
-    "for k = 1 : m-1\n",
+    "plot(x, T(1,:))\n",
+    "axis([0 1 50 200]);\n",
+    "xlabel('Distance'); ylabel('Temperature');\n",
+    "F(1) = getframe(gcf);\n",
+    "\n",
+    "for k = 1 : m - 1\n",
     "    A = zeros(n,n);\n",
     "    b = zeros(n,1);\n",
     "    for i = 1 : n\n",
@@ -276,10 +288,41 @@
     "            b(i) = -T(k,i);\n",
     "        end\n",
     "    end\n",
+    "    \n",
     "    T(k+1, :) = A \\ b;\n",
+    "    plot(x, T(k+1,:))\n",
+    "    axis([0 1 50 200]);\n",
+    "    xlabel('Distance'); ylabel('Temperature');\n",
+    "    F(k+1) = getframe(gcf);\n",
+    "end\n",
+    "close\n",
+    "\n",
+    "%% This generates a GIF of the results (for use in Jupyter Notebook)\n",
+    "filename = 'parabolic_implicit_animated.gif';\n",
+    "for i = 1 : length(F)\n",
+    "    im = frame2im(F(i)); \n",
+    "    [imind,cm] = rgb2ind(im,256); \n",
+    "    % Write to GIF\n",
+    "    if i == 1 \n",
+    "        imwrite(imind,cm,filename,'gif', 'Loopcount',inf, 'DelayTime',1e-3); \n",
+    "    else \n",
+    "        imwrite(imind,cm,filename,'gif','WriteMode','append', 'DelayTime',1e-3); \n",
+    "    end\n",
     "end"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<figure>\n",
+    "  <center>\n",
+    "  <img src=\"parabolic_implicit_animated.gif\" alt=\"movie of implicit parabolic PDE solution\" style=\"width: 500px;\"/>\n",
+    "  <figcaption>Figure: Solution to 1D heat equation with implicit method. Fo = 0.75</figcaption>\n",
+    "  </center>\n",
+    "</figure>"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,