An exact Riemann solver for the Euler equations, with interactive widgets. #riemann-problem #euler-equations

Euler_exact_Riemann_solver.ipynb

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   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "The Riemann problem for the Euler equations"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is an exact Riemann solver for the one-dimensional Euler equations of compressible flow.  It is based on Sections 14.11-14.12 of [Randall LeVeque's finite volume text](http://depts.washington.edu/clawpack/book.html)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/\"><img alt=\"Creative Commons License\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by/4.0/80x15.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">This example</span> by <a href=\"http://davidketcheson.info\">David Ketcheson</a> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/\">Creative Commons Attribution 4.0 International License</a>.  All code examples are also licensed under the [MIT license](http://opensource.org/licenses/MIT)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Contents\n",
      "\n",
      " - <a href=\"#solver\">Riemann solver</a>\n",
      " - <a href=\"#examples\">Example Riemann problems</a>\n",
      "     - <a href=\"#sod\">Sod shock tube</a>\n",
      "     - <a href=\"#expansion\">Expansion</a>\n",
      "     - <a href=\"#collision\">Collision</a>\n",
      " - <a href=\"#interact\">Interactive widget</a>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "from scipy.optimize import fsolve"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "<div id=\"solver\"></div>\n",
      "\n",
      "Riemann solver"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The cell below contains the code for the Riemann solver.  In order to be similar to approximate Riemann solvers used in numerical codes, it takes the conserved quantities *(density, momentum, energy)* as inputs.  For convenience, it returns the solution in primitive variables *(density, velocity, pressure)*."
     ]
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     "input": [
      "def exact_riemann_solution(q_l,q_r,x=None,t=None,gamma=1.4):\n",
      "    \"\"\"Return the exact solution to the Riemann problem with initial states q_l, q_r.\n",
      "       The solution is computed at time t and points x (where x may be a 1D numpy array).\n",
      "       \n",
      "       The input vectors are the conserved quantities but the outputs are [rho,u,p].\n",
      "    \"\"\"\n",
      "    rho_l = q_l[0]\n",
      "    u_l = q_l[1]/q_l[0]\n",
      "    E_l = q_l[2]\n",
      "    \n",
      "    rho_r = q_r[0]\n",
      "    u_r = q_r[1]/q_r[0]\n",
      "    E_r = q_r[2]\n",
      "\n",
      "    # Compute left and right state pressures\n",
      "    p_l = (gamma-1.)*(E_l - 0.5*rho_l*u_l**2)\n",
      "    p_r = (gamma-1.)*(E_r - 0.5*rho_r*u_r**2)\n",
      "\n",
      "    # Compute left and right state sound speeds\n",
      "    c_l = np.sqrt(gamma*p_l/rho_l)\n",
      "    c_r = np.sqrt(gamma*p_r/rho_r)\n",
      "    \n",
      "    alpha = (gamma-1.)/(2.*gamma)\n",
      "    beta = (gamma+1.)/(gamma-1.)\n",
      "\n",
      "    # Check for cavitation\n",
      "    if u_l - u_r + 2*(c_l+c_r)/(gamma-1.) < 0:\n",
      "        print 'Cavitation detected!  Exiting.'\n",
      "        return None\n",
      "    \n",
      "    # Define the integral curves and hugoniot loci\n",
      "    integral_curve_1   = lambda p : u_l + 2*c_l/(gamma-1.)*(1.-(p/p_l)**((gamma-1.)/(2.*gamma)))\n",
      "    integral_curve_3   = lambda p : u_r - 2*c_r/(gamma-1.)*(1.-(p/p_r)**((gamma-1.)/(2.*gamma)))\n",
      "    hugoniot_locus_1 = lambda p : u_l + 2*c_l/np.sqrt(2*gamma*(gamma-1.)) * ((1-p/p_l)/np.sqrt(1+beta*p/p_l))\n",
      "    hugoniot_locus_3 = lambda p : u_r - 2*c_r/np.sqrt(2*gamma*(gamma-1.)) * ((1-p/p_r)/np.sqrt(1+beta*p/p_r))\n",
      "    \n",
      "    # Check whether the 1-wave is a shock or rarefaction\n",
      "    def phi_l(p):        \n",
      "        if p>=p_l: return hugoniot_locus_1(p)\n",
      "        else: return integral_curve_1(p)\n",
      "    \n",
      "    # Check whether the 1-wave is a shock or rarefaction\n",
      "    def phi_r(p):\n",
      "        if p>=p_r: return hugoniot_locus_3(p)\n",
      "        else: return integral_curve_3(p)\n",
      "        \n",
      "    phi = lambda p : phi_l(p)-phi_r(p)\n",
      "\n",
      "    # Compute middle state p, u by finding curve intersection\n",
      "    p,info, ier, msg = fsolve(phi, (p_l+p_r)/2.,full_output=True,xtol=1.e-14)\n",
      "    # For strong rarefactions, sometimes fsolve needs help\n",
      "    if ier!=1:\n",
      "        p,info, ier, msg = fsolve(phi, (p_l+p_r)/2.,full_output=True,factor=0.1,xtol=1.e-10)\n",
      "        # This should not happen:\n",
      "        if ier!=1: \n",
      "            print 'Warning: fsolve did not converge.'\n",
      "            print msg\n",
      "\n",
      "    u = phi_l(p)\n",
      "\n",
      "    \n",
      "    # Find middle state densities\n",
      "    rho_l_star = (p/p_l)**(1./gamma) * rho_l\n",
      "    rho_r_star = (p/p_r)**(1./gamma) * rho_r\n",
      "        \n",
      "    # compute the wave speeds\n",
      "    ws = np.zeros(5) \n",
      "    # The contact speed:\n",
      "    ws[2] = u\n",
      "    \n",
      "    # Find shock and rarefaction speeds\n",
      "    if p>p_l: \n",
      "        ws[0] = (rho_l*u_l - rho_l_star*u)/(rho_l - rho_l_star)\n",
      "        ws[1] = ws[0]\n",
      "    else:\n",
      "        c_l_star = np.sqrt(gamma*p/rho_l_star)\n",
      "        ws[0] = u_l - c_l\n",
      "        ws[1] = u - c_l_star\n",
      "\n",
      "    if p>p_r: \n",
      "        ws[4] = (rho_r*u_r - rho_r_star*u)/(rho_r - rho_r_star)\n",
      "        ws[3] = ws[4]\n",
      "    else:\n",
      "        c_r_star = np.sqrt(gamma*p/rho_r_star)\n",
      "        ws[3] = u+c_r_star\n",
      "        ws[4] = u_r + c_r    \n",
      "    \n",
      "\n",
      "    # Compute return values\n",
      "\n",
      "    # Choose a time based on the wave speeds\n",
      "    if x is None: x = np.linspace(-1.,1.,1000)\n",
      "    if t is None: t = 0.8*max(np.abs(x))/max(np.abs(ws))\n",
      "    \n",
      "    xs = ws*t # Wave locations\n",
      "        \n",
      "    # Find solution inside rarefaction fans\n",
      "    xi = x/t\n",
      "    u1 = ((gamma-1.)*u_l + 2*(c_l + xi))/(gamma+1.)\n",
      "    u3 = ((gamma-1.)*u_r - 2*(c_r - xi))/(gamma+1.)\n",
      "    rho1 = (rho_l**gamma*(u1-xi)**2/(gamma*p_l))**(1./(gamma-1.))\n",
      "    rho3 = (rho_r**gamma*(xi-u3)**2/(gamma*p_r))**(1./(gamma-1.))\n",
      "    p1 = p_l*(rho1/rho_l)**gamma\n",
      "    p3 = p_r*(rho3/rho_r)**gamma\n",
      "    \n",
      "    rho_out = (x<=xs[0])*rho_l + (x>xs[0])*(x<=xs[1])*rho1 + (x>xs[1])*(x<=xs[2])*rho_l_star + (x>xs[2])*(x<=xs[3])*rho_r_star + (x>xs[3])*(x<=xs[4])*rho3 + (x>xs[4])*rho_r\n",
      "    u_out   = (x<=xs[0])*u_l + (x>xs[0])*(x<=xs[1])*u1 + (x>xs[1])*(x<=xs[2])*u + (x>xs[2])*(x<=xs[3])*u + (x>xs[3])*(x<=xs[4])*u3 + (x>xs[4])*u_r\n",
      "    p_out   = (x<=xs[0])*p_l + (x>xs[0])*(x<=xs[1])*p1 + (x>xs[1])*(x<=xs[2])*p + (x>xs[2])*(x<=xs[3])*p + (x>xs[3])*(x<=xs[4])*p3 + (x>xs[4])*p_r\n",
      "    return rho_out, u_out, p_out"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "<div id=\"examples\"></div>\n",
      "\n",
      "Examples"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's try this solver out on some interesting initial states.  Note that the Euler equations are invariant under Galilean transformations, so we can without loss of generality take $u_l+u_r=0$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gamma = 7./5."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_exact_riemann_solution(rho_l=3.,u_l=0.,p_l=3.,rho_r=1.,u_r=0.,p_r=1.,t=0.4):\n",
      "    E_l = p_l/(gamma-1.) + 0.5*rho_l*u_l**2\n",
      "    E_r = p_r/(gamma-1.) + 0.5*rho_r*u_r**2\n",
      "    \n",
      "    q_l = [rho_l, rho_l*u_l, E_l]\n",
      "    q_r = [rho_r, rho_r*u_r, E_r]\n",
      "    \n",
      "    x = np.linspace(-1.,1.,1000)\n",
      "    rho, u, p = exact_riemann_solution(q_l, q_r, x, t, gamma=gamma)\n",
      "\n",
      "    fig = plt.figure(figsize=(18,6))\n",
      "    primitive = [rho,u,p]\n",
      "    names = ['Density','Velocity','Pressure']\n",
      "    axes = [0]*3\n",
      "    for i in range(3):\n",
      "        axes[i] = fig.add_subplot(1,3,i+1)\n",
      "        q = primitive[i]\n",
      "        plt.plot(x,q,linewidth=3)\n",
      "        plt.title(names[i])\n",
      "        qmax = max(q)\n",
      "        qmin = min(q)\n",
      "        qdiff = qmax - qmin\n",
      "        axes[i].set_ylim((qmin-0.1*qdiff,qmax+0.1*qdiff))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div id=\"sod\"></div>\n",
      "\n",
      "#1: Sod shock tube"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we consider the classic shock tube problem, with high density and pressure on the left, low density and pressure on the right.  Both sides are initially at rest.  The solution includes a rarefaction, a contact, and a shock."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_exact_riemann_solution(rho_l=3.,\n",
      "                            u_l = 0.,\n",
      "                            p_l = 3.,\n",
      "                            rho_r=1.,\n",
      "                            u_r = 0.,\n",
      "                            p_r = 1.)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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E7krqz0BBasFAQVKnXnyxuRV21qzidmNSVX7fkernggua4w0b4Kd/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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1071a9310>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "<div id=\"expansion\"></div>\n",
      "\n",
      "2: Symmetric expansion"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next we consider the case of equal densities and pressures, and equal and opposite velocities, with the initial states moving away from each other.  The result is two rarefaction waves (the contact has zero strength)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_exact_riemann_solution(rho_l=1.,\n",
      "                            u_l = -3.,\n",
      "                            p_l = 1.,\n",
      "                            rho_r=1.,\n",
      "                            u_r = 3.,\n",
      "                            p_r = 1.,\n",
      "                            t = 0.1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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w9/s0I1DIIU56yKbly5N65Mh4fQBAGr31lvTcc77u3l26/vq4/QBIL5Y8ZN/a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\nr4kTpQEDfP3ee9L998ftp9JWrJB+/vPkmmleSAQ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avyn82WdmTytM62nm5BlnbJReM6Hw\nXCup2k/u/l7R8TNm9pCZjXX3/XWKsRE07FgiD8dDHh4ycnE85OLaaNixRC6Oh1w8JOTheMjDtZF4\nLNXrlody96q8ImmKmZ1mZiMk/Y2kjXWKKS82SrqycHylQqVsADMbZWajC8fHSfpLSWVX5mwSccbG\nRklXSJKZzZH0btFUuVZRtZ/M7EQzs8LxLElG4jxKK4wl8nB55OHyyMXxkItroxXGErm4PHLx4MjD\n8ZCHayP5WEpxBckLFe6/OCjpbUnPFJ4/WdK/FF23QNLrCqty/m1a8eT1IWmspG5Jb0jaImlMaT9J\n+nOFlUq3S3qtVfppsLEh6TpJ1xVd82Dh/A6VWTm52R/V+knSDYVxs13Sv0mak3XMGfTRE5J+LelQ\nIS9d3QpjiTwcu5/Iw5X7h1xcg34iF5OLycVV+4lcXL5vyMM16CfycDp52ApfBAAAAAAAEFvWuzwA\nAAAAAIAGREEBAAAAAAAkRkEBAAAAAAAkRkEBAAAAAAAkRkEBAAAAAAAkRkEBAAAAAAAkRkEBAAAA\nAAAkRkEBAAAAAAAk9v8fk238fnXXxgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10730b0d0>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Notice that, by symmetry, we must have $u=0$ in the middle state.  As we make the initial velocities larger in magnitude, the rarefaction waves increase in strength until the pressure and density at $x=0$ reach zero.  In the plot above, the middle pressure is very close but not quite equal to zero.\n",
      "\n",
      "For what initial velocity $u_r=-u_l$ does the middle pressure vanish?  We can find it by solving the equation that describes the 3- (or 1-) Riemann invariant with $u=p=0$.  For $\\rho_r=p_r=1$, this gives $u_r \\approx 5.916$.  What happens if the velocities are set to larger than this value in the problem above?\n",
      "\n",
      "More generally, cavitation occurs if\n",
      "\n",
      "$$u_l + 2\\frac{c_l}{\\gamma-1} < u_r + 2\\frac{c_r}{\\gamma-1}.$$\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "<div id=\"collision\"></div>\n",
      "\n",
      "3: Colliding flows"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next, consider the case in which the left and right states are moving toward eachother.  This leads to a pair of shocks, with a high-density, high-pressure state in between."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_exact_riemann_solution(rho_l=1.,\n",
      "                            u_l = 3.,\n",
      "                            p_l = 1.,\n",
      "                            rho_r=1.,\n",
      "                            u_r = -3.,\n",
      "                            p_r = 1.,\n",
      "                            t = 0.4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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Ok2WA/uw0AOhLT0xiaMDUbDOmOFyXRhWgP7UYWAc98WLq8PYyNAAABknDzhBV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\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10730be50>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "<div id=\"interact\"></div>\n",
      "\n",
      "Interactive Riemann solver"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here you can set up your own Riemann problem and immediately see the solution.  If you don't want to download and run the notebook, an online interactive version is [here](http://sagecell.sagemath.org/?z=eJytWNtu47YWfQ-Qf2BnHiLJsmzFGeAgqIsCnfaxOCgGpw-DwJAtOiaOLgwviTJf30VSlKjYSgdog8FEIdfeXPvKLbGat0KRRtf8lRSSNPz6irm1ulC8alXF9hl_NU9mn1fq-uoo2prIA-OvWcsVq9k3Snqho2yrZ3p9dX1V0iOhXXFQO8FoXTTNDltasbaJnnZV-rQTabf9vW1oqtyvx6Kui22e3cX311cEPx8-fPiDKi0aok7U6SJeB1GtXf3D6SZctPuK1uSFqRNhDVOsqIhUhaKSmOPwn8icWvx8geSgiUlyaGuuFS1JoQjsoQS2NiXhLWuUJB2JXk5UUDzUxSvZU1KQ_LP3mRDFazyqnpzBGqglz_SgWiEBpZbzoW0kFc847kkXjQJXkNwDaDZbrSDjwF_FqU11yh-ywSHuAeu7imyNZV_XD25NDyv5wyrc-HXYuO1XRiXC7ohAiV-xSkSgxG8YJW7tI_nF-Y1U9Og8JtjjSTm_IyRUSo3_HJpbGpGN8jLP4iQyxJZknX1KrD0JLEiS29jDxRlcBHABuHDw72IjW40VySkte0IHS6jhmXwSyh2UgOTKkok9RlzACIsRcejNouKnYsJ4Fd1mif2rB-6pGhELixjAgRUnevg_ObaCHIpnBu7IUbfHjjbISxulBblNIpiwAMVQEfmRrO-HJEQQkMLk5pdBFSmpQj7S8gcEtUPuNY_ZzYgXruBMQYbWfSSf6ZE1Ln-hkT4K1NdBI4uldfVJP7YNQ4-o2gPr6fawnYXtcixtSVXU-7IgnNxbW4wVMGIVxjnPlhFfIRJxkkQX3RnHF0_YnJ9gMsacIC6eIP72BG_WDmZpCRtmLfA5cus0JKPamCQkivKls2kA5guTD4lbvXzcZt6c7ztOXDxOxPE0ti7l0OMQXWFDnC9fimdqWmNB5KnFLvJRoCcd0YSHhDQtnp_Yrop4fO8zaMwl5Cv_aQvz7n1avXUn5EY4rSQdkG-TZ0D-W5yF4XzOVcxx3Xw31ymy72YnNo2ld9vSczlvYzUry4r6ZpoSTfavBEVYomRd6dmThaSBeTxlzbFNCaMiJbV8xKnuSo5wUEoiRGNh8-I2S4-6qnbuvtl-EZqmnWorXMB0md_Fns1vcKJUosWZgS9lioZaU3NXyv4A0pjWSk604kO7AosftnnYjv4JPXN2K7brLA-YrgNff8Sli_Aj-LoqSYNudCo4p800zj2pIFGDRvlnIRr4994bVTKnCHf2MxWPNOyVoxgs8fHTMMkH168NvkTwpmEtaSPt_R_c6zts2btvaIJ5tnJtCaVtIcEFPkGLS2jxNhk_-nnHlo0tmvBefJHuyvtGRSujT_Eg9cXNLsrOYUbg3gtgKICMvmCtK0R7GY_ZMznOVp5tEiNPaFwbjdEwGKDxjc5JNHq1m4HC5XiiIDcKrCK3bEt2RBxGR7-53ldzGi0lx-XgYxAept2GlfSh760Tb6y7G6yzc0xvhphaN12ectn01t3NWSfes25Wo154yXO6buowE9EQ5POm1bfE56LSJqnHbt220szNdrreFxLDb9tczj-4rDNd3Mwg93i0JlSskbw40AhXXGr-rde-8oFXI14Bv87-k9RFF0Gu2Muoi-NV8OeLnN5_nbSuTBR4_mnIoOHbUUmeF47L6eHNAcVb0klmHwsv1pmO361U3xbM6BDMGomfHcwMh4eOxX6KW9hx0AptzoTcBBAd7MNFIcQ3H0vHjwc6XwKNeXk1Trpx3y2C2WFQsRnz06vo2FJvpirErApuOOCMxCjJ-5m6V9UjNhYhLGLTT9QTxNjlcAsYPt2P286UYexeGOC2qPupX-k3c7eZD3v5uHfrBV1peMTtiNj0CDFFbEbEnUNshr07L-JfoAxXMmWr57nqWaZ6jqCe46WnrLTnxC9x4vOc-CwnPseJz3HiU07cc-r7RB_c1PktdVSB8Algp7WqxTuyyRLAUjMJdX5yOzIzQfBKZXjCq2aEX5J9o9soX6d3Qy4KVuOSRWVvx1dqt9UUZoDB8s1nexO_3qQ3_6PmHcY-_rd_hb3p4UXn0OuHZNMzwHjE0AjQBJpHGm3CmdKgvzLTOkErK8pyJ_XeWBPl6SZlizxowE_GDs8TQsHIBOusUJc-pWiD9IWV6rTdxFMIXFTRyNoD8VAzWh-Umwb4NFkGaSyzZrpcsuPRvOkbqaVFnRmUSap2rxWro8jsLzGQJVYuNVKL4U_vf_u5CAMyxqKD9B-KcGHUv9mgOdD4twni0KBtOwCfPLszt8n11c925kW3HZNjN_OZydb6Vlbo0QIu_5SayRFCha4U_JfqcXf56c32Ok75u8K25uf2c6NcvK_8PWFAxs9g7vsNn7wlo6jefjcJv9PwyQtvAPZfTcIG-2T1f7W6UjKoTM3BDx4iPEQ4iFVkIGLyTenvb-uhjIF85_OgORKFnhI0BfdhMPySEtQE9A2LQ59Agadd_BeW1ABm&lang=python)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import display, HTML\n",
      "from IPython.html import widgets\n",
      "from IPython.html.widgets import interact, interactive"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "interact(plot_exact_riemann_solution,\n",
      "         rho_l=(1.,10.,0.1),\n",
      "         u_l=(-5.,5.,0.1),\n",
      "         p_l=(1.,10.,0.1),\n",
      "         rho_r=(1.,10.,0.1),\n",
      "         u_r=(-5.,5.,0.1),\n",
      "         p_r=(1.,10.,0.1),\n",
      "         t=(0.1,1.,0.1));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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qA/3jgAOkn/u5bLxtW5ZUANAdoyYUbM+UNFfSP0ga7nQ7T9LVkhQRiyUdbJv7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\nExeyANB7XItXEwkFoASoUAC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       "text": [
        "<matplotlib.figure.Figure at 0x107c34f90>"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}

There is a bug in this exact Riemann solver. When computing rho_l_star and rho_r_star you need to use

Find middle state densities

if (p<=p_l):
    rho_l_star = (p/p_l)**(1.0/gamma) * rho_l
else:
    rho_l_star = ((1.0+beta*p/p_l)/((p/p_l)+beta))*rho_l;

if (p<=p_r):
    rho_r_star = (p/p_r)**(1.0/gamma) * rho_r
else:
    rho_r_star = ((1.0+beta*p/p_r)/((p/p_r)+beta))*rho_r;