Computational_Seismology_Igel/dg_scalar_advection_1d.ipynb

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    "<div style='background-image: url(\"../../share/images/header.svg\") ; padding: 0px ; background-size: cover ; border-radius: 5px ; height: 250px'>\n",
    "    <div style=\"float: right ; margin: 50px ; padding: 20px ; background: rgba(255 , 255 , 255 , 0.7) ; width: 50% ; height: 150px\">\n",
    "        <div style=\"position: relative ; top: 50% ; transform: translatey(-50%)\">\n",
    "            <div style=\"font-size: xx-large ; font-weight: 900 ; color: rgba(0 , 0 , 0 , 0.8) ; line-height: 100%\">Computational Seismology</div>\n",
    "            <div style=\"font-size: large ; padding-top: 20px ; color: rgba(0 , 0 , 0 , 0.5)\">The (nodal) Discontinuous Galerkin Method for the linear advection equation</div>\n",
    "        </div>\n",
    "    </div>\n",
    "</div>"
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    "<p style=\"width:20%;float:right;padding-left:50px\">\n",
    "<img src=../../share/images/book.jpg>\n",
    "<span style=\"font-size:smaller\">\n",
    "</span>\n",
    "</p>\n",
    "\n",
    "\n",
    "---\n",
    "\n",
    "This notebook is part of the supplementary material \n",
    "to [Computational Seismology: A Practical Introduction](https://global.oup.com/academic/product/computational-seismology-9780198717416?cc=de&lang=en&#), \n",
    "Oxford University Press, 2016.\n",
    "\n",
    "\n",
    "##### Authors:\n",
    "* Stephanie Wollherr ([@swollherr](https://github.com/swollherr))\n",
    "* Heiner Igel ([@heinerigel](https://github.com/heinerigel))\n",
    "---"
   ]
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    "The following notebook presents a 1D discontinuous Galerkin code for the advection equation using two different approaches for the integration in time.\n",
    "\n",
    "---"
   ]
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    "### The discretization of the advection equation\n",
    "\n",
    "To keep things simple we want to solve the linear advection equation as a (scalar) hyperbolic equation:\n",
    "\n",
    "$$ \\partial_t u + \\mu \\ \\partial_x u=0$$\n",
    "1. We first derive the **weak form** of the equation by multiplying it by an arbitrary test function and integration by parts.\n",
    "\n",
    "2. As in the Spectral Element Method, we approximate our unknown variable u(x,t) by a sum over space-dependent basis functions $\\psi_i$ weighted by time-dependent coefficients $u_i(t)$\n",
    "\n",
    "$$ u(x,t) \\ \\approx \\ \\overline{u}(x,t) \\ = \\ \\sum_{i=1}^{n} u_i(t) \\\n",
    "\\psi_i(x) $$\n",
    "\n",
    "As interpolating functions we choose the **Lagrange polynomials** and use\n",
    "$\\xi$ as the space variable representing our elemental domain:\n",
    "\n",
    "$$ \\psi_i \\ \\rightarrow \\ \\ell_i^{(N)} (\\xi) \\ := \\ \\prod_{k = 1, \\ k\n",
    "\\neq i}^{N+1} \\frac{\\xi - \\xi_k}{\\xi_i-\\xi_k}, \\qquad   i = 1, 2, \\dotsc\n",
    ", N + 1  $$\n",
    "\n",
    "Finally we get the (local) semi-discrete scheme for the k-th element that we derived in the lecture:\n",
    "\n",
    "$$  M^k \\cdot \\partial_{t} u_h^k(t) -  \\mu \\ (S^k)^T \\cdot u_h^k(t)= -[(\\mu \\ u)^*(t) \\ l_j^k(x)]_{x^{k}_l}^{x^{k}_r}$$\n",
    "\n",
    "<div style=\"clear:both\"></div> \n",
    "\n",
    "---\n",
    "\n",
    "We will use the following ingredients that we already discussed in previous exercises during the spectral element method:\n",
    "* Interpolating polynomials (**the Lagrange polynomials**, calculated in the routine 'lagrange.py'  )\n",
    "* Numerical integration (**the Gauss-Lobatto-Legendre approach**, getting the GLL points $\\xi_{i}$ and their corresponding weights in the routine 'gll.py')\n",
    "* How we can calculate the **mass and stiffness matrix** using some nice properties of our basis functions.\n",
    "\n"
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    "## Exercises ##\n",
    "\n",
    "### 1. Examine the code below ###\n",
    "Look how the mass and stiffness matrix are calculated and compare them to the elemental system matrices for the spectral element code. What are the differences? <br>\n",
    "Take a closer look at the stiffness matrix: where is the Jacobian that we use to map our element to the reference element $I=[-1,1]$?  <br>\n",
    "How is the flux calculated at the boundaries? Try different values for $\\alpha \\in [0,1]$ and see what happens.\n",
    "What kind of flux do we get for $\\alpha=1$ or $\\alpha=0$? <br>\n",
    "Try different orders of approximations.\n",
    "Try the Euler and the Runge-Kutta scheme for the time extrapolation. What do you notice?\n",
    "\n",
    "### 2. Time schemes ###\n",
    "\n",
    "Formulate the analytical solution to the advection problem and plot it along with the numerical solution each time step you visualize it during extrapolation.\n",
    "Formulate an error between analytical and numerical result. \n",
    "Analyze the solution error as a function of propagation distance for the Euler scheme and the predictor-corrector scheme. <br>\n",
    "(You find the implementation already in the code using exercice = 2).\n",
    "\n",
    "### 3. The Courant Criterion ###\n",
    "Using the sampling code below find numerically the Courant limit (for a fixed time extrapolation to $t_{max}$), keeping all other parameters constant, increasing the global spatial order $N$ of the scheme.<br>\n",
    "(like in the SEM Code).\n",
    "\n",
    "### 4. How discontinuous is the discontinuous Galerkin method? ###\n",
    "Extract the field values at the element boundaries and calculate the relative amount of the field discontinuity. <br>\n",
    "How do the discontinuities compare with the flux values?\n",
    "(You find the implementation already in the code using exercice = 4).\n",
    "\n",
    "<div style=\"clear:both\"></div>\n",
    "\n",
    "---"
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    "### The Code "
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    "# This is a configuration step. Please run it before the simulation code!\n",
    "\n",
    "# Imports etc\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "# Show Plot in The Notebook\n",
    "matplotlib.use(\"nbagg\")\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# import of DG modules from same directory as the notebook\n",
    "# we use the same functions that we already know from the spectral element method\n",
    "from gll import gll\n",
    "from lagrange import lagrange\n",
    "from lagrange1st import lagrange1st\n",
    "from legendre import legendre"
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    "# ---------------------------------------------------------------\n",
    "#\n",
    "# Discontinous galerkin method for the\n",
    "# 1D advection equation using an upwind flux\n",
    "#\n",
    "\n",
    "# ---------------------------------------------------------------\n",
    "# Initialization of setup\n",
    "# ---------------------------------------------------------------\n",
    "nt = 800        # number of time steps   \n",
    "xmax = 30.      # Length of domain\n",
    "rho = 1         # Density for homogeneous model, if you change the density, take that into account for a analytical solution!\n",
    "mu = 20         # Speed of the wave, positive=propagating to the right, negative=propagating to the left \n",
    "N = 6           # Order of Lagrange polynomials\n",
    "ne = 100        # Number of elements\n",
    "\n",
    "\n",
    "# ----------------------------------------------------------------\n",
    "\n",
    "# Initialization of GLL points integration weights\n",
    "[xi, w] = gll(N)  # xi -> N+1 coordinates [-1 1] of GLL points\n",
    "                  # w integration weights at GLL locations\n",
    "    \n",
    "# Space domain\n",
    "le = xmax / ne  # Length of elements for regular grid points\n",
    "\n",
    "\n",
    "# Vector with GLL points, same values for boundary points\n",
    "k = -1\n",
    "xg = np.zeros((N +1)* ne)\n",
    "#xg[k] = 0\n",
    "for i in range(1, ne + 1):\n",
    "    for j in range(0, N+1):\n",
    "        k += 1\n",
    "        xg[k] = (i - 1) * le + .5 * (xi[j] + 1) * le\n",
    "\n",
    "x=np.reshape(xg,(ne,(N+1))) #Matrix containing all GLL points in every element\n",
    "x=x.T\n",
    "\n",
    "\n",
    "# ---------------------------------------------------------------\n",
    "# Calculation if time step according to Courant criterion\n",
    "\n",
    "dxmin = min(np.diff(xg[0:N]))\n",
    "eps = 0.1    # Courant value orig 0.1\n",
    "\n",
    "dt = eps * dxmin / (np.abs(mu))  # Global time step\n",
    "\n",
    "#-----------------------------------------------------------------\n",
    "#---Exercice 3: Find the courant limit, dt fixed and N decreasing\n",
    "#dt=0.0004 #or some other max. timestep\n",
    "#eps= dt*(np.abs(mu)/dxmin)\n",
    "#print(eps)\n",
    "#----------------------------------------------------------------\n",
    "\n",
    "\n",
    "J = le / 2  # Mapping - Jacobian, same for every element if the element size is everywhere the same\n",
    "Ji = 1 / J  # Inverse Jacobian\n",
    "\n",
    "# Initialization of 1st derivative of Lagrange polynomials used in the stiffness matrix\n",
    "l1d = lagrange1st(N)  # Array with GLL as columns for each N+1 polynomial\n"
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    "### The system matrices ###\n",
    "The mass and the stiffness matrix can be precalculated and stored when we transform every element to a reference elemen. <br>\n",
    "In the 1D case this is simply the transformation to the interval $I=[-1,1]$ with $dx= J \\ d\\xi$ and hence $J=\\frac{dx}{d\\xi}$. Additionally we also have to transform a spacial derivative with respect to $x$ into a spacial derivative with respect to $\\xi$. <br>\n",
    "The Lagrange polynomials have some nice properties that we will use to calculate the mass and stiffness matrix.\n",
    "At the GLL-points $x_j$ it holds:\n",
    "$$ l_i(x_j) = \\delta_{ij}= \\begin{cases} 1 \\ \\text{ if $i=j$} \\\\ 0 \\ \\text{ if $i \\neq j $} \\end{cases} $$\n",
    "For the numerical integration we use the numerical quadrature, that uses the GLL and their corresponding weights to approximate an integral.\n",
    "\n",
    "\n",
    "$$ M_{ij}^k=\\int_{-1}^1 l_i^k(\\xi)  l_j^k(\\xi) \\ J \\ d\\xi = \\sum_{m=1}^{N_p} w_m \\ l_i^k (x_m)  l_j^k(x_m)\\ J =\\sum_{m=1}^{N_p} w_m \\delta_{im}\\ \\delta_{jm} \\ J= \\begin{cases} w_i \\ J \\ \\ \\text{ if } i=j \\\\ 0 \\ \\ \\ \\ \\ \\ \\ \\text{   if } i \\neq j\\end{cases}$$\n",
    " **Note** : We have a diagonal mass matrix!\n",
    " \n",
    " \n",
    " $$ S_{i,j}= \\int_{-1}^1 l_i^k(\\xi) \\cdot \\partial _x  l_j^k(\\xi) \\ d\\xi= \\sum_{m=1}^{N_p} w_m \\ l_i^k(x_m)\\cdot \\partial_x l_j^k(x_m)= \\sum_{m=1}^{N_p} w_m \\delta_{im}\\cdot \\partial_xl_j^k(x_m)= w_i \\cdot \\partial_x l_j^k(x_i) $$\n",
    "We already calculated the Lagrange polynomials. They will be used to calculate the first derivatives of the Lagrange polynomials in the function \"Lagrange1st\". Now we have all the ingredients to calculate the mass and stiffness matrix:"
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    "# -----------------------------------------------------------------\n",
    "# Initialization of system matrices\n",
    "# -----------------------------------------------------------------\n",
    "\n",
    "\n",
    "# variables for elemental matrices\n",
    "Me = np.zeros(N + 1, dtype=float)\n",
    "Ke = np.zeros([N + 1, N + 1], dtype=float)\n",
    "\n",
    "\n",
    "# Mass matrix\n",
    "# Elemental\n",
    "for i in range(-1, N):\n",
    "    Me[i + 1] = rho * w[i + 1] * J #only a vector since the matrix is diagonal\n",
    "\n",
    "\n",
    "\n",
    "# Build inverse matrix\n",
    "Minv = np.identity(N+1)\n",
    "for i in range(0, N+1):\n",
    "    Minv[i,i] = 1. / Me[i]\n",
    "\n",
    "\n",
    "# ---------------------------------------------------------------\n",
    "# Stiffness Matrix\n",
    "# Elemental\n",
    "\n",
    "for i in range(-1, N):\n",
    "    for j in range(-1, N):\n",
    "            Ke[i+1,j+1] = mu * w[j + 1] * l1d[i + 1, j + 1] #corrected indices\n",
    "\n",
    "\n",
    "# ---------------------------------------------------------------"
   ]
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    "### The numerical flux\n",
    "\n",
    "The big difference to the Spectral Element Method is the communication between the element neighbors using a flux term $(\\mu \\ u)^*$. <br>\n",
    "For the 1D advection equation we can derive the general formula for a numerical upwind flux\n",
    "$$ (\\mu \\ u )^*= \\frac{\\mu}{2}(u^++u^-) +\\frac{|\\mu|}{2}(\\hat n^+u^++\\hat n^-u^-)$$\n",
    "with $u^-$ as the interior information, \n",
    "$u^+$ the exterior information and \n",
    "$\\hat n$ the corresponding outer pointing normal vectors. <br>\n",
    "This leads to the update on the right boundary of an element\n",
    "$$ \\frac{\\mu}{2}\\big(u_h^k(x_r^k)+u_h^{k+1}(x_l^{k+1})\\big )+ \\frac{|\\mu|}{2}\\big(u_h^k(x_r^k)-u_h^{k+1}(x_l^{k+1})\\big )$$\n",
    "and on the left boundary of an element\n",
    "$$\\frac{\\mu}{2}\\big(u_h^k(x_l^k)+u_h^{k-1}(x_r^{k-1}) \\big)+ \\frac{|\\mu|}{2}(-u_h^{k}(x_l^{k}+u_h^{k-1}(x_r^{k-1}))\\big)$$\n",
    "\n",
    "For a positive travel speed $\\mu$ (-> $\\mu=|\\mu|$) the value on the left boundary is the value of the previous element $k-1$ from where the wave is coming from. On the right boundary it's just the value of the actual element $k$ (and vice versa for $\\mu$ being negative).\n",
    "Notice that we only consider the first (left boundary) and the last (right boundary) DOF of an element since the Lagrange polynomials are zero for every other points.\n",
    "This is calculated by the function \"flux\":"
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    "#calculates the flux between two boundary sides of connected elements for\n",
    "#element i\n",
    "#Flux matrix du\n",
    "def flux(alpha, u, N, ne, mu): #What kind of flux do we get for alpha=1 or alpha=0?\n",
    "    \n",
    "#impose boundary conditions at x=0 and x=end\n",
    "    ubd1 = 0 \n",
    "    ubd2 = 0 \n",
    "        \n",
    "    du = np.zeros((N+1, ne)) # for every element we have 2 faces to other elements (left and right)\n",
    "    for i in range(0, ne):\n",
    "    \n",
    "\n",
    "        if i==0: # left boundary of the domain\n",
    "            du[0,i] = -mu / 2 * (u[0,i] + ubd1) - (1 - alpha) * abs(mu) / 2 * (ubd1 - u[0,i]) #left flux\n",
    "            du[N,i] = mu / 2 * (u[N,i] + u[0,i+1]) + (1 - alpha) * abs(mu) / 2 * (u[N,i] - u[0,i+1])#right flux\n",
    "\n",
    "        elif i==ne-1:  # right boundary of the domain\n",
    "            du[0,i] = -mu / 2 * (u[0,i] + u[N,i-1]) - (1-alpha) * abs(mu) / 2 * (-u[0,i] + u[N,i-1])\n",
    "            du[N,i] = mu/2 * (u[N,i] + ubd2) + (1-alpha) * abs(mu) / 2 * (u[N,i] - ubd2)\n",
    "            \n",
    "        else: # in the middle of the domain\n",
    "            du[0,i] = -mu / 2 * (u[0,i] + u[N,i-1]) - (1-alpha)*abs(mu) / 2 * (-u[0,i] + u[N,i-1])\n",
    "            du[N,i] = +mu / 2 * (u[N,i] + u[0,i+1]) + (1-alpha)*abs(mu) / 2 * (u[N,i] - u[0,i+1])\n",
    "\n",
    "\n",
    "    return du\n",
    "    return udiff"
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    "### The time extrapolation\n",
    "\n",
    "Last but not least we have to solve our semidiscrete scheme that we derived above using an appropriate time extrapolation:\n",
    "\\begin{eqnarray*}\n",
    "\\partial_{t} u_h^k(t)= \\underbrace{(M^k)^{-1} \\left( \\mu \\ (S^k) \\cdot u_h^k(t) -\\text{flux}^k \\right )}_{\\text{RHS(u(t))}}\n",
    "\\end{eqnarray*}\n",
    "**Note:** If we have a homogenuous medium and a constant advection velocity for every element we can also formulate the equation above as matrix-vector products since $M^k$ and $S^k$ are the same for every element $k$. \n",
    "In the code below we implemented two different time extrapolation schemes:\n",
    "\n",
    "1. The Euler scheme in time with the update\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "u(t+1)= u(t) + dt \\cdot \\text{RHS(u(t))}\n",
    "\\end{eqnarray*}\n",
    "\n",
    "2. and a second-order Runge-Kutta method (also called predictor-corrector scheme) with the follwing steps\n",
    "\n",
    "\\begin{eqnarray*} \n",
    "k1(t)&=&RHS(u(t)) \\\\\n",
    "k2(t)&=&RHS(u(t)+dt\\cdot k1(t)) \\\\\n",
    "& & \\\\\n",
    "u(t+1)&=& u(t) + 0.5 \\cdot dt \\cdot \\left(k1(t)+k2(t)\\right)\n",
    "\\end{eqnarray*}\n",
    "\n",
    "In the Euler method we use for example a loop over every element, in the Runge-Kutta method we use the matrix-vector formulation. <br>"
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       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        mouse_event_fn(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
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   "source": [
    "##------------------------------------------------------\n",
    "# Initialize solution vectors\n",
    "u = np.zeros(((N+1), ne))\n",
    "uold = u\n",
    "unew = u\n",
    "\n",
    "x0 = 10 #center of the impulse\n",
    "uold = 0.5 * np.exp(-0.4 * (x - x0) ** 2) #initial condition/impulse\n",
    "\n",
    "#############\n",
    "#choose between the different exercises\n",
    "exercise = 1\n",
    "#############\n",
    "\n",
    "if exercise==2: #exercise 2, error between analytical and numerical solution\n",
    "   ua = np.zeros(((N+1),ne))\n",
    "   error = np.zeros((nt+1,(N+1)*ne)) #error array such that we can plot the error as a function of propagation distance\n",
    "   ua = uold\n",
    "    \n",
    "elif exercise==4:# exercise 4: how discontinuous is the dg method?\n",
    "    udiff = np.zeros((nt+1,ne))\n",
    "\n",
    "    \n",
    "############### Time extrapolation ####################################\n",
    "#\n",
    "# Choose between the Euler scheme and the second order Runge-Kutta scheme\n",
    "method = 1 #Euler=0, RK=1\n",
    "iplot = 10 # output interval\n",
    "\n",
    "u = uold\n",
    "\n",
    "\n",
    "#initialization for different plots, depending on the exercise\n",
    "if exercise==2:\n",
    "    uplot = np.reshape(u,((N+1)*ne),1) #u is reshaped to a vector in order to plot it against xg=vector with all the gll points\n",
    "    uaplot = np.reshape(ua,((N+1)*ne),1)\n",
    "    \n",
    "    fig = plt.figure()\n",
    "    ax1 = fig.add_subplot(2,1,1)\n",
    "    ax2 = fig.add_subplot(2,1,2)\n",
    "\n",
    "    line, = ax1.plot(xg,uplot,color=\"blue\",lw=1.5)\n",
    "    line2, = ax1.plot(xg,uplot,color=\"red\",lw=0.5)\n",
    "    \n",
    "    line3, = ax2.plot(error[0,:], color='green', lw=1.5)\n",
    "\n",
    "    plt.ylim(0,0.7)\n",
    "    plt.xlabel(' x (m)')\n",
    "    plt.ylabel(' Amplitude ')\n",
    "    fig.show();\n",
    "       \n",
    "    \n",
    "elif exercise==1 or exercise==3:\n",
    "    uplot = np.reshape(u,((N+1)*ne),1)\n",
    "    fig = plt.figure()\n",
    "    line, = plt.plot(xg,uplot,color=\"blue\",lw=1.5)\n",
    "    plt.ylim(0,0.7)\n",
    "    plt.xlabel(' x (m)')\n",
    "    plt.ylabel(' Amplitude ')\n",
    "    fig.show();\n",
    "\n",
    "alpha = 1 #you can choose alpha between [0,1] to get different flux concepts\n",
    "\n",
    "\n",
    "####################\n",
    "##-------Time loop\n",
    "\n",
    "for it in range(1, nt+1):\n",
    "    \n",
    "    if method==0: #Euler scheme\n",
    "        Flux=flux(alpha,u,N,ne,mu) #calculates the flux at the boundaries of an element\n",
    "        \n",
    "        # Extrapolation for every element\n",
    "        for k in range(0,ne):\n",
    "               unew[:,k] = dt* (np.dot(Minv, np.dot(Ke, u[:,k]) - Flux[:,k])) + u[:,k]\n",
    "        u = unew\n",
    "\n",
    "    \n",
    "    else: #RK scheme\n",
    "            Flux = flux(alpha,u,N,ne,mu) #calculates the flux at the boundaries of an element,\n",
    "            k1 = np.dot(Minv, np.dot(Ke, u) - Flux)\n",
    "            Flux = flux(alpha, u + dt * k1, N, ne, mu) #for a second order scheme this needs to be done twice in every iteration\n",
    "            k2 = np.dot(Minv, np.dot(Ke, u+dt*k1)-Flux)\n",
    "            u = u + 0.5 * dt * (k1 + k2)\n",
    "    \n",
    "##--Plotting section--###\n",
    "\n",
    "    if not it % iplot:\n",
    "        if exercise==2: #plot that compares the analytical and numerical solution, calculates the error\n",
    "            uplot = np.reshape(u,((N+1) * ne),1)\n",
    "\n",
    "            ua = 0.5*np.exp(-0.4*(x - (x0 + it * dt * mu)) ** 2)#analytical solution of the advection equation\n",
    "            uaplot = np.reshape(ua,((N+1) * ne),1)\n",
    "        \n",
    "            line.set_ydata(uplot)\n",
    "            line2.set_ydata(uaplot)\n",
    "            error[it,:] = np.abs(uplot - uaplot)\n",
    "            #maxerror=np.max(np.abs(uplot-uaplot))\n",
    "            #error[it,:]=np.linalg.norm(uplot-uaplot) #2Norm\n",
    "\n",
    "            ax1.set_title('Time step : %g ' %it)\n",
    "            ax1.set_ylim([0.,0.7])\n",
    "            fig.canvas.draw()\n",
    "        \n",
    "            lyaxis = np.max(np.abs(error)) #length of y-axis\n",
    "            ax2.set_ylim([0,lyaxis])\n",
    "    \n",
    "            line3.set_ydata(error[it,:])\n",
    "            distance = it * dt * mu\n",
    "            ax2.set_title('Error (numerical-analytical) solution')\n",
    "\n",
    "        \n",
    "        elif exercise==4: #calculate the differences at the boundary points between two elements in every time step\n",
    "            udiff[it,0:ne-1] = np.abs(u[0,1:ne] - u[N,0:ne-1])\n",
    "        \n",
    "        \n",
    "        else: #plots only the solution u, for exercises 1,3\n",
    "            uplot=np.reshape(u,((N+1)*ne),1)\n",
    "            line.set_ydata(uplot)\n",
    "            plt.title('Time step : %g ' %it)\n",
    "            fig.canvas.draw()\n",
    "        \n",
    "### end of time loop        \n",
    "####################\n",
    "\n",
    "\n",
    "## Exercise 4: visualize the difference at the boundaries of an element at a given iteration step it\n",
    "if exercise==4:\n",
    "    itplot = input('Give a iteration step between 1 and %g: ' %nt)\n",
    "\n",
    "    fig2=plt.figure()\n",
    "    line4 = plt.plot(udiff[itplot,:], color='green', marker='o',label='grid points')    \n",
    "    #ax = plt.axes()    \n",
    "\n",
    "    \n",
    "    fig2.show()\n",
    "    lyaxis=np.max(udiff[itplot,:])\n",
    "    plt.ylim(0,lyaxis)\n",
    "\n",
    "    plt.title('Difference at the boundaries at iteration step %g' %itplot)\n",
    "    plt.legend(loc=2)\n",
    "    fig2.canvas.draw()\n"
   ]
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