Computational_Seismology_Igel/dg_elastic_hetero_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)\">Discontinuous Galerkin Method - 1D Elastic Wave Equation, Heterogeneous case</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",
    "* David Vargas ([@dvargas](https://github.com/davofis))\n",
    "* Heiner Igel ([@heinerigel](https://github.com/heinerigel))"
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    "## Basic Equations\n",
    "\n",
    "The source-free elastic wave equation in 1D reads \n",
    "\n",
    "\\begin{align}\n",
    "\\partial_t \\sigma - \\mu \\partial_x v & = 0 \\\\ \n",
    "\\partial_t v - \\frac{1}{\\rho} \\partial_x \\sigma & = 0\n",
    "\\end{align}\n",
    "\n",
    "with $\\rho$ the density and $\\mu$ the shear modulus. This equation in matrix-vector notation follows\n",
    "\n",
    "\\begin{equation}\n",
    "\\partial_t \\mathbf{Q} + \\mathbf{A} \\partial_x \\mathbf{Q} = 0\n",
    "\\end{equation}\n",
    "\n",
    "where $\\mathbf{Q} = (\\sigma, v)$ is the vector of unknowns and the matrix $\\mathbf{A}$ contains the parameters $\\rho$ and $\\mu$. We seek to solve the linear advection equation as a hyperbolic equation $ \\partial_t u + \\mu \\ \\partial_x u=0$. A series of steps need to be done:\n",
    "\n",
    "1) The weak form of the equation is derived by multiplying both sides by an arbitrary test function. \n",
    "\n",
    "2) Apply the stress Free Boundary Condition after integration by parts\n",
    "\n",
    "3) We approximate the unknown field $\\mathbf{Q}(x,t)$ by a sum over space-dependent basis functions $\\ell_i$ weighted by time-dependent coefficients $\\mathbf{Q}(x_i,t)$, as we did in the spectral elements method. As interpolating functions we choose the Lagrange polynomials and use $\\xi$ as the space variable representing the elemental domain:\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{Q}(\\xi,t) \\ = \\ \\sum_{i=1}^{N_p} \\mathbf{Q}(\\xi_i,t) \\ell_i(\\xi) \\qquad with \\qquad \\ell_i^{(N)} (\\xi) \\ := \\ \\prod_{j = 1, \\ j \\neq i}^{N+1} \\frac{\\xi - \\xi_j}{\\xi_i-\\xi_j}, \\quad i,j = 1, 2, \\dotsc , N + 1 \n",
    "\\end{equation}\n",
    "\n",
    "4) The continuous weak form is written as a system of linear equations by considering the approximated displacement field. Finally, the semi-discrete scheme can be written in matrix-vector form as\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{M}\\partial_t \\mathbf{Q} = \\mathbf{A}\\mathbf{K}\\mathbf{Q} - \\mathbf{Flux}\n",
    "\\end{equation}\n",
    "\n",
    "5) Time extrapolation is done after applying a standard 1st order finite-difference approximation to the time derivative, we call it the Euler scheme.\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{Q}^{t+1} \\approx \\mathbf{Q}^{t} + dt\\mathbf{M}^{-1}(\\mathbf{A}\\mathbf{K}\\mathbf{Q} - \\mathbf{Flux})\n",
    "\\end{equation}\n",
    "\n",
    "This notebook implements both Euler and Runge-Kutta schemes for solving the free source version of the elastic wave equation in a homogeneous media. To keep the problem simple, we use as spatial initial condition a Gauss function with half-width $\\sigma$\n",
    "\n",
    "\\begin{equation}\n",
    "Q(x,t=0)  = e^{-1/\\sigma^2 (x - x_{o})^2}\n",
    "\\end{equation}"
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    "# Import all necessary libraries, this is a configuration step for the exercise.\n",
    "# Please run it before the simulation code!\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",
    "from gll import gll\n",
    "from lagrange1st import lagrange1st\n",
    "from flux_hetero import flux"
   ]
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   "source": [
    "### 1. Initialization of setup"
   ]
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    "# Initialization of setup\n",
    "# --------------------------------------------------------------------------\n",
    "tmax   = 2.5          # Length of seismogram [s]\n",
    "xmax   = 10000        # Length of domain [m]\n",
    "vs0    = 2500         # Advection velocity\n",
    "rho0   = 2500         # Density [kg/m^3]\n",
    "mu0    = rho0*vs0**2  # shear modulus\n",
    "N      = 2            # Order of Lagrange polynomials\n",
    "ne     = 200          # Number of elements\n",
    "sig    = 100          # width of Gaussian initial condition\n",
    "x0     = 4000         # x locartion of Gauss\n",
    "eps    = 0.2          # Courant criterion\n",
    "iplot  = 20           # Plotting frequency\n",
    "imethod = 'RK'        # 'Euler', 'RK'\n",
    "\n",
    "nx = ne*N + 1\n",
    "dx = xmax/(nx-1)      # space increment\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",
    "# Space domain\n",
    "le = xmax/ne       # Length of elements, here equidistent\n",
    "ng = ne*N + 1\n",
    "\n",
    "# Vector with GLL points\n",
    "k = 0\n",
    "xg = np.zeros((N+1)*ne)\n",
    "for i in range(0, ne):\n",
    "    for j in range(0, N+1):\n",
    "        k += 1\n",
    "        xg[k-1] = i*le + .5*(xi[j] + 1)*le\n",
    "\n",
    "x = np.reshape(xg, (N+1, ne), order='F').T\n",
    "\n",
    "# Calculation of time step acoording to Courant criterion\n",
    "dxmin = np.min(np.diff(xg[1:N+1]))\n",
    "dt = eps*dxmin/vs0 # Global time step\n",
    "nt = int(np.floor(tmax/dt))\n",
    "\n",
    "# Mapping - Jacobian\n",
    "J = le/2  # Jacobian\n",
    "Ji = 1/J  # Inverse Jacobian\n",
    "\n",
    "# 1st derivative of Lagrange polynomials\n",
    "l1d = lagrange1st(N)"
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    "### 2. Elemental Mass and Stiffness matrices\n",
    "\n",
    "The mass and the stiffness matrix are calculated prior time extrapolation, so they are pre-calculated and stored at the beginning of the code. \n",
    "\n",
    "The integrals defined in the mass and stiffness matrices are computed using a numerical quadrature, in this cases the GLL quadrature that uses the GLL points and their corresponding weights to approximate the integrals. Hence,\n",
    "\n",
    "\\begin{equation}\n",
    "M_{ij}^k=\\int_{-1}^1 \\ell_i^k(\\xi)  \\ell_j^k(\\xi) \\ J \\ d\\xi = \\sum_{m=1}^{N_p} w_m \\ \\ell_i^k (x_m)  \\ell_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",
    "\\end{equation}\n",
    "\n",
    "that is a **diagonal mass matrix!**. Subsequently, the stiffness matrices is given as \n",
    " \n",
    "\\begin{equation} \n",
    "K_{i,j}= \\int_{-1}^1 \\ell_i^k(\\xi) \\cdot \\partial _x  \\ell_j^k(\\xi) \\ d\\xi= \\sum_{m=1}^{N_p} w_m \\ \\ell_i^k(x_m)\\cdot \\partial_x \\ell_j^k(x_m)= \\sum_{m=1}^{N_p} w_m \\delta_{im}\\cdot \\partial_x\\ell_j^k(x_m)= w_i \\cdot \\partial_x \\ell_j^k(x_i) \n",
    "\\end{equation}\n",
    "\n",
    "The Lagrange polynomials and their properties have been already used, they determine the integration weights $w_i$ that are returned by the python method \"gll\". Additionally, the fist derivatives of such basis, $\\partial_x \\ell_j^k(x_i)$, are needed, the python method \"Lagrange1st\" returns them."
   ]
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    "# Initialization of system matrices\n",
    "# -----------------------------------------------------------------\n",
    "# Elemental Mass matrix\n",
    "M = np.zeros((N+1, N+1))\n",
    "for i in range(0, N+1):\n",
    "    M[i, i] = w[i] * J\n",
    "\n",
    "# Inverse matrix of M (M is diagonal!)\n",
    "Minv = np.identity(N+1)\n",
    "for i in range(0, N+1):\n",
    "    Minv[i,i] = 1. / M[i,i]\n",
    "\n",
    "# Elemental Stiffness Matrix\n",
    "K = np.zeros((N+1, N+1))\n",
    "for i in range(0, N+1):\n",
    "    for j in range(0, N+1):\n",
    "            K[i,j] = w[j] * l1d[i,j] # NxN matrix for every element\n"
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    "### 3. Flux Matrices\n",
    "\n",
    "The main difference in the heterogeneous case with respect the homogeneous one is found in the definition of fluxes. As in the case of finite volumes when we solve the 1D elastic wave equation, we allow the coefficients of matrix A to vary inside the element.\n",
    "\n",
    "\\begin{equation} \n",
    "\\mathbf{A}=\n",
    "     \\begin{pmatrix}\n",
    "       0 & -\\mu_i \\\\\n",
    "       -1/\\rho_i & 0 \n",
    "     \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Now we need to diagonalize $\\mathbf{A}$. Introducing the seismic impedance $Z_i = \\rho_i c_i$, we have\n",
    " \n",
    "\\begin{equation}\n",
    "\\mathbf{A} = \\mathbf{R}^{-1}\\mathbf{\\Lambda}\\mathbf{R}\n",
    "\\qquad\\text{,}\\qquad\n",
    "\\mathbf{\\Lambda}=\n",
    "     \\begin{pmatrix}\n",
    "       -c_i & 0 \\\\\n",
    "        0 & c_i \n",
    "     \\end{pmatrix}\n",
    "\\qquad\\text{,}\\qquad\n",
    "\\mathbf{R} =  \n",
    "     \\begin{pmatrix}\n",
    "       Z_i & -Z_i \\\\\n",
    "       1 &  1 \n",
    "     \\end{pmatrix}\n",
    "\\qquad\\text{and}\\qquad\n",
    "\\mathbf{R}^{-1} = \\frac{1}{2Z_i}\n",
    "     \\begin{pmatrix}\n",
    "       1 & Z_i \\\\\n",
    "      -1 & Z_i \n",
    "     \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "We decompose the solution into right propagating $\\mathbf{\\Lambda}^{+}$ and left propagating eigenvalues $\\mathbf{\\Lambda}^{-}$ where\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{\\Lambda}^{+}=\n",
    "     \\begin{pmatrix}\n",
    "       -c_i & 0 \\\\\n",
    "        0 & 0 \n",
    "     \\end{pmatrix}\n",
    "\\qquad\\text{,}\\qquad\n",
    "\\mathbf{\\Lambda}^{-}=\n",
    "     \\begin{pmatrix}\n",
    "        0 & 0 \\\\\n",
    "        0 & c_i \n",
    "     \\end{pmatrix}\n",
    "\\qquad\\text{and}\\qquad\n",
    "\\mathbf{A}^{\\pm} = \\mathbf{R}^{-1}\\mathbf{\\Lambda}^{\\pm}\\mathbf{R}\n",
    "\\end{equation}\n",
    "\n",
    "This strategy allows us to formulate the Flux term in the discontinuous Galerkin method. The following cell initializes all flux related matrices"
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    "# Inialize Flux relates matrices\n",
    "# ---------------------------------------------------------------\n",
    "\n",
    "# initialize heterogeneous A\n",
    "Ap  = np.zeros((ne,2,2))\n",
    "Am  = np.zeros((ne,2,2))\n",
    "Z   = np.zeros(ne)\n",
    "rho = np.zeros(ne)\n",
    "mu  = np.zeros(ne)\n",
    "\n",
    "# initialize c, rho, mu, and Z\n",
    "rho = rho + rho0\n",
    "rho[int(ne/2):ne] = .25 * rho[int(ne/2):ne] # Introduce discontinuity \n",
    "mu = mu + mu0\n",
    "c = np.sqrt(mu/rho)\n",
    "Z = rho * c\n",
    "\n",
    "# Initialize flux matrices\n",
    "for i in range(1,ne-1):\n",
    "    # Left side positive direction\n",
    "    R = np.array([[Z[i], -Z[i]], [1, 1]])\n",
    "    Lp = np.array([[0, 0], [0, c[i]]])\n",
    "    Ap[i,:,:] = R @ Lp @ np.linalg.inv(R)\n",
    "\n",
    "    # Right side negative direction\n",
    "    R = np.array([[Z[i], -Z[i]], [1, 1]])\n",
    "    Lm = np.array([[-c[i], 0 ], [0, 0]])\n",
    "    Am[i,:,:] = R @ Lm @ np.linalg.inv(R)"
   ]
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    "### 4. Discontinuous Galerkin Solution\n",
    "\n",
    "The principal characteristic of the discontinuous Galerkin Method is the communication between the element neighbors using a flux term, in general it is given \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{Flux} = \\int_{\\partial D_k} \\mathbf{A}\\mathbf{Q}\\ell_j(\\xi)\\mathbf{n}d\\xi\n",
    "\\end{equation}\n",
    "\n",
    "this term leads to four flux contributions for left and right sides of the elements\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{Flux} = -\\mathbf{A}_{k}^{-}\\mathbf{Q}_{l}^{k}\\mathbf{F}^{l} + \\mathbf{A}_{k}^{+}\\mathbf{Q}_{r}^{k}\\mathbf{F}^{r} - \\mathbf{A}_{k}^{+}\\mathbf{Q}_{r}^{k-1}\\mathbf{F}^{l} + \\mathbf{A}_{k}^{-}\\mathbf{Q}_{l}^{k+1}\\mathbf{F}^{r}\n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "\n",
    "Last but not least, we have to solve our semi-discrete scheme that we derived above using an appropriate time extrapolation, in the code below we implemented two different time extrapolation schemes:\n",
    "\n",
    "1) **Euler scheme** \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{Q}^{t+1} \\approx \\mathbf{Q}^{t} + dt\\mathbf{M}^{-1}(\\mathbf{A}\\mathbf{K}\\mathbf{Q} - \\mathbf{Flux})\n",
    "\\end{equation}\n",
    "\n",
    "2) Second-order **Runge-Kutta method** (also called predictor-corrector scheme) \n",
    "\n",
    "\\begin{eqnarray*} \n",
    "k_1 &=& f(t_i, y_i) \\\\\n",
    "k_2 &=& f(t_i + dt, y_i + dt k_1) \\\\\n",
    "& & \\\\\n",
    "y_{i+1} &=& y_i + \\frac{dt}{2} (k_1 + k_2)\n",
    "\\end{eqnarray*}\n",
    "\n",
    "with $f$ that corresponds with $\\mathbf{M}^{-1}(\\mathbf{A}\\mathbf{K}\\mathbf{Q} - \\mathbf{Flux})$"
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       "        },\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"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<div id='3f384d89-22f8-43a0-9a2e-42dff18d5315'></div>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# DG Solution, Time extrapolation\n",
    "# ---------------------------------------------------------------\n",
    "\n",
    "# Initalize solution vectors\n",
    "Q    = np.zeros((ne, N+1, 2))\n",
    "Qnew = np.zeros((ne, N+1, 2))\n",
    "\n",
    "k1 = np.zeros((ne, N+1, 2))\n",
    "k2 = np.zeros((ne, N+1, 2))\n",
    "\n",
    "Q[:,:,0] = np.exp(-1/sig**2*((x-x0))**2)\n",
    "Qs = np.zeros(xg.size)  # for plotting\n",
    "Qv = np.zeros(xg.size)  # for plotting\n",
    "\n",
    "# Initialize animated plot\n",
    "# ---------------------------------------------------------------\n",
    "fig = plt.figure(figsize=(10,6))\n",
    "ax1 = fig.add_subplot(2,1,1)\n",
    "ax2 = fig.add_subplot(2,1,2)\n",
    "line1 = ax1.plot(x, Q[:,:,0], 'k', lw=1.5)\n",
    "line2 = ax2.plot(x, Q[:,:,1], 'r', lw=1.5)\n",
    "ax1.axvspan(((nx-1)/2+1)*dx, nx*dx, alpha=0.2, facecolor='b')\n",
    "ax2.axvspan(((nx-1)/2+1)*dx, nx*dx, alpha=0.2, facecolor='b')\n",
    "ax1.set_xlim([0, xmax])\n",
    "ax2.set_xlim([0, xmax])\n",
    "\n",
    "ax1.set_ylabel('Stress')\n",
    "ax2.set_ylabel('Velocity')\n",
    "ax2.set_xlabel(' x ')\n",
    "plt.suptitle('Heterogeneous Disc. Galerkin - %s method'%imethod, size=16)\n",
    "\n",
    "plt.ion() # set interective mode\n",
    "plt.show()\n",
    "\n",
    "# ---------------------------------------------------------------\n",
    "# Time extrapolation\n",
    "# ---------------------------------------------------------------\n",
    "for it in range(nt):\n",
    "    if imethod == 'Euler':\n",
    "        # Calculate Fluxes\n",
    "        Flux = flux(Q, N, ne, Ap, Am)\n",
    "        for i in range(1,ne-1):\n",
    "            Qnew[i,:,0] = dt * Minv @ (-mu[i] * K @ Q[i,:,1].T - Flux[i,:,0].T) + Q[i,:,0].T\n",
    "            Qnew[i,:,1] = dt * Minv @ (-1/rho[i] * K @ Q[i,:,0].T - Flux[i,:,1].T) + Q[i,:,1].T\n",
    "\n",
    "    elif imethod == 'RK':\n",
    "        # Calculate Fluxes\n",
    "        Flux = flux(Q, N, ne, Ap, Am)\n",
    "        for i in range(1,ne-1):\n",
    "            k1[i,:,0] = Minv @ (-mu[i] * K @ Q[i,:,1].T - Flux[i,:,0].T)\n",
    "            k1[i,:,1] = Minv @ (-1/rho[i] * K @ Q[i,:,0].T - Flux[i,:,1].T)\n",
    "\n",
    "        for i in range(1,ne-1):\n",
    "            Qnew[i,:,0] = dt * Minv @ (-mu[i] * K @ Q[i,:,1].T - Flux[i,:,0].T) + Q[i,:,0].T\n",
    "            Qnew[i,:,1] = dt * Minv @ (-1/rho[i] * K @ Q[i,:,0].T - Flux[i,:,1].T) + Q[i,:,1].T\n",
    "\n",
    "        Flux = flux(Qnew,N,ne,Ap,Am)\n",
    "        \n",
    "        for i in range(1,ne-1):\n",
    "            k2[i,:,0] = Minv @ (-mu[i] * K @ Qnew[i,:,1].T - Flux[i,:,0].T)\n",
    "            k2[i,:,1] = Minv @ (-1/rho[i] * K @ Qnew[i,:,0].T - Flux[i,:,1].T)\n",
    "            \n",
    "        # Extrapolate\n",
    "        Qnew = Q + 0.5 * dt * (k1 + k2)\n",
    "\n",
    "    else:\n",
    "        raise NotImplementedError\n",
    "\n",
    "    Q, Qnew = Qnew, Q\n",
    "\n",
    "    # --------------------------------------   \n",
    "    # Animation plot. Display solution  \n",
    "    if not it % iplot:\n",
    "        for l in line1:\n",
    "            l.remove()\n",
    "            del l\n",
    "        for l in line2:\n",
    "            l.remove()\n",
    "            del l\n",
    "\n",
    "        # stretch for plotting\n",
    "        k = 0\n",
    "        for i in range(ne):\n",
    "            for j in range(N+1):\n",
    "                Qs[k] = Q[i,j,0]\n",
    "                Qv[k] = Q[i,j,1]\n",
    "                k = k + 1\n",
    "        # -------------------------------------- \n",
    "        # Display lines\n",
    "        line1 = ax1.plot(xg, Qs, 'k', lw=1.5)\n",
    "        line2 = ax2.plot(xg, Qv, 'r', lw=1.5)\n",
    "        plt.gcf().canvas.draw()\n"
   ]
  }
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