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   "source": [
    "# PDE session 1\n",
    "## Finite differences using python"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this session we will use python to compute the derivative of a dependent variable $u$ as function of an independent variable $x$.\n",
    "\n",
    "The dependent variable $u$ could be interpreted as the temperature, a displacement or any other scalar variable of interest. In this python notebook we are concerned with how this variable evolves along a 1D bar of length 2 meters. We will compute the gradient of $u$ using backward, forward and centered finite differences.\n",
    "\n",
    "The independent variable $x$, the dependent variable $u$ and a reference derivative $du/dx$ are stored in a text file. The reference derivative will be used to determine the accuracy of the various finite difference schemes.\n",
    "\n",
    "You should carry the following tasks in this notebook:\n",
    "\n",
    "0. Import the necessary python modules\n",
    "1. Import the data needed in this session\n",
    "2. Split the data into the independent variable $x$, the dependent variable $u$ and its derivative $du/dx$ using array slicing\n",
    "3. Plot the dependent variable $u$ and its derivative $du/dx$ using a subplot (i.e. one plot per quantity)\n",
    "4. Write a function, called backward, which computes $du/dx$ using a backward finite difference where the inputs are the dependent variable $u$ and the independent variable $x$\n",
    "5. Write a function, called forward, which computes $du/dx$ using a forward finite difference where the inputs are the dependent variable $u$ and the independent variable $x$\n",
    "6. Write a function, called centered, which computes $du/dx$ using a centered finite difference where the inputs are the dependent variable $u$ and the independent variable $x$\n",
    "7. Plot the derivatives computed using the functions or questions 4,5 and 6 and plot them alongside the reference data in a subplot (i.e. one plot per method)\n",
    "8. What can you comment upon the results plotted in question 7?\n",
    "9. Write a function, called abs_error, which computes the absolute error between the reference derivative $du/dx$ and its numerical estimation\n",
    "10. Plot the errors obtained with the three different finite difference approximations using a subplot (i.e. one plot per method), what can you say about your results?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 0**\n",
    "\n",
    "We first start by importing the necessary modules for our notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Question 1**\n",
    "\n",
    "Next, we need to import our dataset which is stored in the file named `data_1.csv`. This file should be stored in the same folder as this python notebook.\n",
    "\n",
    "To load this data file we use the package `loadtxt` from `numpy`. We will look a bit more closely to the import of data in a dedicated notebook. For the time being we will skip the first row of the text file and use a , as a delimeter. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = np.loadtxt('data_1.csv',delimiter=',',skiprows=1)"
   ]
  }
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