Merge branch 'main' of github.com:rhwhite/numeric_2024 into main

worksheets/Worksheets_Lab3/Worksheet_UG_Lab3_2_2024.ipynb

@@ -0,0 +1,157 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "1533bd8e",
+   "metadata": {},
+   "source": [
+    "Names: "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "8be05185",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# your import statements\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c17b4ce4",
+   "metadata": {},
+   "source": [
+    "Consider a long hallway (length $L$) in an office building. If we assume that any\n",
+    "cigarette smoke mixes across the width of the hallway and vertically\n",
+    "through the depth of the hallway much faster than it mixes along the\n",
+    "hallway, we can write the diffusion of cigarette smoke as an equation\n",
+    "$$\\frac {\\partial S} {\\partial t}\n",
+    "= \\frac {\\partial \\kappa \\partial S}{\\partial x^2}$$\n",
+    "where $S$ is the concentration of smoke, $\\kappa$ is the rate of diffusion of\n",
+    "smoke, t is the time and x is\n",
+    "distance along the hallway.  At the centre of the hallway is the smoker and here the smoke is 5 su (smoke units).  At one end of the hallway (x=0) is an open window, here the smoke is 0. At the other end of the hallway (x=L) is a closed window (no smoke through the closed window)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9eb3d006",
+   "metadata": {},
+   "source": [
+    "**Question 1**\n",
+    "\n",
+    "Under what conditions can you move $\\kappa$ outside the derivative?  Make that assumption here, and also make the steady state assumption.  What is your new differential equation?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5f110569",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ace733e0",
+   "metadata": {},
+   "source": [
+    "**Question 2**\n",
+    "\n",
+    "Using a centre-difference scheme, separating your hallway into $N=10$ divisions (so $N+1=11$ grid points) write down linear equations for the 1th to $N/2-1=4$th grid points and the $N/2+1=6$th to $N-1=9$th grid points. Note that the 0th and $N=10$th grid points are boundary points and there is a special condition at the $N/2=5$th grid point."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b0f3282e",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6bcb3435",
+   "metadata": {},
+   "source": [
+    "**Question 3**\n",
+    "\n",
+    "What type of boundary condition is at $x=0$ (0th grid point)?  Write it as linear equation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1abcc6cf",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "305884ee",
+   "metadata": {},
+   "source": [
+    "**Question 4**\n",
+    "\n",
+    "What type of boundary condition is at $x=L$ (N=10th grid point)?  Write it as linear equation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9f3f791b",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3ffd3078",
+   "metadata": {},
+   "source": [
+    "**Question 5**\n",
+    "\n",
+    "Consider the condition at the centre of the hallway ($x=L/2$, $N/2=5$th grid point) Write it as linear equation."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "648910b3",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "68450247",
+   "metadata": {},
+   "source": [
+    "**Question 6**\n",
+    "\n",
+    "Put all your equations into a matrix equation and solve the matrix equation.  Plot the solution."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "577152a7",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "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": "3.10.1"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}