Add solution to problem in finite difference deriv module, courtesy B…

…randon Clark.

Tutorial-Finite_Difference_Derivatives-FDtable_soln.ipynb

@@ -0,0 +1,99 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Creating the Finite-Difference Table for Centered First and Second  Derivatives, from Second through Eighth-Order Accuracy\n",
+    "\n",
+    "## Courtesy Brandon Clark"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Derivative Accuracy   -4     -3     -2   -1     0     1    2     3     4   \n",
+      "---------- -------- ------ ------ ----- ---- ------- --- ----- ----- ------\n",
+      "     First        2      0      0     0 -1/2       0 1/2     0     0      0\n",
+      "     First        4      0      0  1/12 -2/3       0 2/3 -1/12     0      0\n",
+      "     First        6      0  -1/60  3/20 -3/4       0 3/4 -3/20  1/60      0\n",
+      "     First        8  1/280 -4/105   1/5 -4/5       0 4/5  -1/5 4/105 -1/280\n",
+      "    Second        2      0      0     0    1      -2   1     0     0      0\n",
+      "    Second        4      0      0 -1/12  4/3    -5/2 4/3 -1/12     0      0\n",
+      "    Second        6      0   1/90 -3/20  3/2  -49/18 3/2 -3/20  1/90      0\n",
+      "    Second        8 -1/560  8/315  -1/5  8/5 -205/72 8/5  -1/5 8/315 -1/560\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Step 0: Import needed modules\n",
+    "import numpy as np\n",
+    "import finite_difference as fin\n",
+    "from astropy.table import Table\n",
+    "\n",
+    "# Step 1: Set the maximum finite-difference accuracy order\n",
+    "max_fdorder = 8\n",
+    "\n",
+    "# Step 2: Set up table parameters\n",
+    "#    One column for deriv order, one for deriv accuracy, and max_fdorder+1\n",
+    "numcols = 2 + max_fdorder + 1\n",
+    "#    8 rows: 4 accuracy orders per derivative order.\n",
+    "numrows = 8\n",
+    "#    Center column index of table will be at 2 + max_fdorder/2  (zero-offset indexing)\n",
+    "column_corresponding_to_zero_fd_point = 2 + int(max_fdorder/2)\n",
+    "#    The table is initialized as a matrix of zeroes in numpy:\n",
+    "numpy_matrix = np.zeros((numrows, numcols), dtype=object)\n",
+    "\n",
+    "# Step 3: Construct the first-order derivative finite difference coefficients\n",
+    "rowcount = 0\n",
+    "for fdorder in range(2, max_fdorder+1, 2): # loop runs from 2 to max_fdorder inclusive, skipping odd orders.\n",
+    "    numpy_matrix[rowcount, 0] = \"First\"\n",
+    "    numpy_matrix[rowcount, 1] = fdorder     \n",
+    "    fdcoeffs, fdstencl = fin.compute_fdcoeffs_fdstencl(\"D0\", fdorder)\n",
+    "    for i in range(fdorder):\n",
+    "        numpy_matrix[rowcount, column_corresponding_to_zero_fd_point + fdstencl[i][0]] = fdcoeffs[i]\n",
+    "    rowcount += 1\n",
+    "\n",
+    "# Step 4: Construct the first-order derivative finite difference coefficients\n",
+    "for fdorder in range(2, max_fdorder+1, 2): # loop runs from 2 to max_fdorder inclusive, skipping odd orders.\n",
+    "    numpy_matrix[rowcount, 0] = \"Second\"\n",
+    "    numpy_matrix[rowcount, 1] = fdorder\n",
+    "    fdcoeffs, fdstencl = fin.compute_fdcoeffs_fdstencl(\"DD00\", fdorder)\n",
+    "    for i in range(fdorder+1):\n",
+    "        numpy_matrix[rowcount, column_corresponding_to_zero_fd_point + fdstencl[i][0]] = fdcoeffs[i]\n",
+    "    rowcount += 1  \n",
+    "\n",
+    "# Step 5: Construct an astropy table from the numpy matrix with the following header info, and then print it:\n",
+    "table = Table(numpy_matrix, names=('Derivative', 'Accuracy', '-4', '-3', '-2', '-1', '0', '1', '2', '3', '4'))\n",
+    "print(table)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.6.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

Tutorial-Finite_Difference_Derivatives.ipynb

@@ -99,7 +99,7 @@
     "\n",
     "In addition, if the gridfunction $u$ exists on a grid that is less than four (spatial) dimensions, it is up to the user to truncate the additional index information.\n",
     "\n",
-    "#### Exercise: Using compute_fdcoeffs_fdstencl(), write the necessary loops to output the finite difference coefficient tables in the Wikipedia article on [finite difference coefficients](https://en.wikipedia.org/wiki/Finite_difference_coefficients), for up to second-order derivatives (i.e., up to $\\partial_i^2$), and for upwinded derivatives up to second-order accuracy."
+    "#### Exercise: Using compute_fdcoeffs_fdstencl(), write the necessary loops to output the finite difference coefficient tables in the Wikipedia article on [finite difference coefficients](https://en.wikipedia.org/wiki/Finite_difference_coefficients), for first and second centered derivatives (i.e., up to $\\partial_i^2$)  up to eighth-order accuracy. [Solution, courtesy Brandon Clark](Tutorial-Finite_Difference_Derivatives-FDtable_soln.ipynb)."
    ]
   },
   {