diff --git a/Guide_to_limiters.ipynb b/Guide_to_limiters.ipynb
index 92f8bbe..1d3a104 100644
--- a/Guide_to_limiters.ipynb
+++ b/Guide_to_limiters.ipynb
@@ -11,7 +11,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "#An illustrated guide to limiters\n",
+    "# An illustrated guide to limiters\n",
     "## Or: how to interpolate non-smooth data without creating wiggles"
    ]
   },
@@ -26,7 +26,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "##Table of contents\n",
+    "## Table of contents\n",
     "- [Motivation: interpolation and wiggles](#Interpolation-and-wiggles)\n",
     "- [The simplest limiter: Minmod](#The-simplest-limiter:-Minmod)\n",
     "- [Other TVD limiters](#TVD-limiters)\n",
@@ -36,9 +36,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "%matplotlib inline\n",
@@ -51,9 +49,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "import mpld3             # Skip this cell if you don't have mpld3 installed\n",
@@ -65,7 +61,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "##Interpolation and wiggles"
+    "## Interpolation and wiggles"
    ]
   },
   {
@@ -78,9 +74,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "k = 5\n",
@@ -103,9 +97,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def piecewise_constant_interp(x,y,xx):\n",
@@ -118,7 +110,6 @@
     "yy = piecewise_constant_interp(x,y,xx)\n",
     "plt.figure(figsize=size)\n",
     "plt.plot(xx,yy,'-k',lw=2)\n",
-    "plt.hold(True)\n",
     "plt.plot(x,y,'or',markersize=10,alpha=0.5)\n",
     "plt.axis( (-k, k, -0.1, 2.1) );\n",
     "plt.title('Piecewise-constant approximation',fontsize=20);"
@@ -146,9 +137,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def piecewise_linear_interp(x,y,xx, fd='centered'):\n",
@@ -170,7 +159,6 @@
     "    for i, fd in enumerate( ('centered','forward','backward') ):\n",
     "        yy = piecewise_linear_interp(x,y,xx,fd=fd)\n",
     "        ax[i].plot(xx,yy,'-k',lw=2)\n",
-    "        ax[i].hold(True)\n",
     "        ax[i].plot(x,y,'or',markersize=10,alpha=0.5)\n",
     "        ax[i].axis( axis );\n",
     "        ax[i].text(.5,.9,fd,\n",
@@ -206,9 +194,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "y = np.sin(x/2.)+1. + 2.*(x>0)\n",
@@ -243,9 +229,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def pw_minmod(x,y,xx):\n",
@@ -268,15 +252,12 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "yy = pw_minmod(x,y,xx)\n",
     "plt.figure(figsize=size)\n",
     "plt.plot(xx,yy,'-k',lw=2)\n",
-    "plt.hold(True)\n",
     "plt.plot(x,y,'or',markersize=10,alpha=0.5)\n",
     "plt.axis( (-4,4,-0.5,4.8) );\n",
     "plt.title('Minmod approximation',fontsize=20);"
@@ -292,9 +273,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "from matplotlib.patches import Rectangle\n",
@@ -303,7 +282,6 @@
     "yy = pw_minmod(x,y,xx)\n",
     "plt.figure(figsize=(width,6))\n",
     "plt.plot(xx,yy,'-k',lw=2)\n",
-    "plt.hold(True)\n",
     "plt.plot(x,y,'or',markersize=10,alpha=0.5)\n",
     "plt.axis( (-4,4,-0.1,4.1) );\n",
     "plt.title('minmod approximation',fontsize=20);\n",
@@ -314,7 +292,7 @@
     "        currentAxis = plt.gca()\n",
     "        currentAxis.add_patch(Rectangle((x_avgs[0], y_avgs[0]), \n",
     "                                         x_avgs[1]-x_avgs[0], y_avgs[1]-y_avgs[0], \n",
-    "                                         facecolor=\"grey\",alpha=0.2))"
+    "                                         facecolor=\"grey\",alpha=0.5))"
    ]
   },
   {
@@ -328,7 +306,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "##Total variation"
+    "## Total variation"
    ]
   },
   {
@@ -371,9 +349,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def phi(theta,limiter):\n",
@@ -408,9 +384,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "y = np.sin(x/2.)+1. + 2.*(x>0)\n",
@@ -418,7 +392,6 @@
     "for i, limiter in enumerate( ('minmod', 'vanleer','superbee','MC') ):\n",
     "    yy = pw_limited(x,y,xx,limiter=limiter)\n",
     "    ax[i].plot(xx,yy,'-k',lw=2)\n",
-    "    ax[i].hold(True)\n",
     "    ax[i].plot(x,y,'or',markersize=10,alpha=0.5)\n",
     "    ax[i].axis( (-4,4,-0.1,4.4) );\n",
     "    ax[i].text(.8,.2,limiter,\n",
@@ -439,19 +412,17 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "#Interactive comparison\n",
+    "# Interactive comparison\n",
     "In each region, these limiter take three data points and give back a linear interpolant.  It's illuminating to compare their behavior on a single set of 3 points.  Note that the interactive plot below doesn't work on nbviewer; you'll need to download and run the notebook yourself."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
-    "from IPython.html.widgets import interact, FloatSlider, RadioButtons\n",
+    "from ipywidgets import interact, FloatSlider, RadioButtons\n",
     "from IPython.display import display\n",
     "%matplotlib inline"
    ]
@@ -459,9 +430,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "xx = np.linspace(-0.5,0.5)\n",
@@ -477,18 +446,16 @@
     "        theta = y3\n",
     "    else:\n",
     "        theta = y3/(-y1)\n",
-    "    ax.hold(True)\n",
     "    forward_slope = y3\n",
     "    backward_slope = -y1\n",
     "    plt.fill_between(xx,xx*forward_slope,xx*backward_slope,color='k',alpha=0.2,zorder=0)\n",
     "    for limiter in ('minmod', 'vanleer','superbee','MC'):\n",
     "        sigma = phi(np.array(theta),limiter)*(-y1)\n",
     "        ax.plot(xx,sigma*xx,alpha=0.5,lw=2)\n",
-    "    ax.legend( ('minmod', 'vanleer','superbee','MC'), loc='best' )\n",
+    "    ax.legend( ('','minmod', 'vanleer','superbee','MC'), loc='best' )\n",
     "    ax.plot(x,y,'ok',markersize=15,alpha=0.5)\n",
-    "    ax.hold(False)\n",
     "\n",
-    "    return fig\n",
+    "    #return fig\n",
     "\n",
     "interact(compare_limiters,y1=FloatSlider(min=-1., max=1., step=0.1, value=-0.3,description='$y_{i-1}$',labelcolor='k'),#,orientation='vertical'),\n",
     "                          y3=FloatSlider(min=-1., max=1., step=0.1, value=0.8,description='$y_{i+1}$'));#,orientation='vertical'));"
@@ -520,9 +487,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "# Note: uses PyWENO v. 0.11.2\n",
@@ -533,9 +498,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "import mpld3             # Skip this cell if you don't have mpld3 installed\n",
@@ -548,9 +511,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "weno_order = 5  # must be odd\n",
@@ -568,9 +529,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def stencil_interpolant(x,y,n,offset):\n",
@@ -584,12 +543,10 @@
     "        fig, axis = plt.subplots(figsize=size)\n",
     "    xc = np.linspace(-0.5,0.5)\n",
     "    xx = np.linspace(-(k-1),k-1)\n",
-    "    plt.hold(True)\n",
     "    for i, interpolant in enumerate(interpolants):\n",
     "        axis.plot(xx,interpolant(xx),'-'+color[i])\n",
     "        axis.plot(xc,interpolant(xc),'-'+color[i],linewidth=5,alpha=0.5)\n",
     "    axis.plot(x,y,'ok')\n",
-    "    axis.hold(False)\n",
     "    axis.axis((-(k-.5),k-.5,-0.5,2.1));"
    ]
   },
@@ -610,9 +567,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "p_opt = stencil_interpolant(x,y,5,0)\n",
@@ -623,15 +578,13 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "fig, ax = plt.subplots(3,1,figsize=(width,10))\n",
     "names = ['left','right','center']\n",
     "p = []\n",
-    "for i in range(k):\n",
+    "for i in range(int(k)):\n",
     "    p.append(stencil_interpolant(x,y,k,i))\n",
     "    plot_interpolants(x,y,[p[i]],axis=ax[i],color=[colors[i]])\n",
     "    ax[i].set_title(names[i]+' interpolant')"
@@ -647,9 +600,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "plot_interpolants(x,y,p+[p_opt],color='brgk')"
@@ -673,9 +624,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def compute_opt_weights(k,xi):\n",
@@ -719,9 +668,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def compute_smoothness_indicators(y,k):\n",
@@ -753,9 +700,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "def compute_weights(gamma, beta, epsilon=1.e-6):\n",
@@ -771,9 +716,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "q = {}\n",
@@ -792,9 +735,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "plot_interpolants(x,y,p+[p_opt],color=['b','r','g','k'])\n",
@@ -815,9 +756,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "styles = { 'left' : 'b', 'center' : 'r', 'right' : 'g'}\n",
@@ -878,9 +817,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "%matplotlib inline\n",
@@ -899,9 +836,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "y=np.array( (1,1,1,0,0) )\n",
@@ -934,9 +869,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "from clawpack import pyclaw\n",
@@ -1000,9 +933,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": false
-   },
+   "metadata": {},
    "outputs": [],
    "source": [
     "#This cell may take a few seconds to run\n",
@@ -1067,32 +998,43 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "metadata": {
-    "collapsed": true
-   },
+   "metadata": {},
    "outputs": [],
    "source": []
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 2",
+   "display_name": "3.9",
    "language": "python",
-   "name": "python2"
+   "name": "3.9"
   },
   "language_info": {
    "codemirror_mode": {
     "name": "ipython",
-    "version": 2
+    "version": 3
    },
    "file_extension": ".py",
    "mimetype": "text/x-python",
    "name": "python",
    "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython2",
-   "version": "2.7.9"
+   "pygments_lexer": "ipython3",
+   "version": "3.9.10"
+  },
+  "toc": {
+   "base_numbering": 1,
+   "nav_menu": {},
+   "number_sections": true,
+   "sideBar": true,
+   "skip_h1_title": false,
+   "title_cell": "Table of Contents",
+   "title_sidebar": "Contents",
+   "toc_cell": false,
+   "toc_position": {},
+   "toc_section_display": true,
+   "toc_window_display": false
   }
  },
  "nbformat": 4,
- "nbformat_minor": 0
+ "nbformat_minor": 1
 }
diff --git a/Lesson_03_High-resolution_methods.ipynb b/Lesson_03_High-resolution_methods.ipynb
index f848e47..6ca64e0 100644
--- a/Lesson_03_High-resolution_methods.ipynb
+++ b/Lesson_03_High-resolution_methods.ipynb
@@ -212,7 +212,7 @@
    "source": [
     "We'll use these interface values to approximate the flux, based on the **Lax-Friedrichs flux**:\n",
     "\n",
-    "$$F_\\imh = \\frac{1}{2} \\left( f(q^-_\\imh) + f(q^+_\\imh) - \\frac{\\Dt}{\\Dx} (q^+_\\imh - q^-_\\imh)\\right)$$\n",
+    "$$F_\\imh = \\frac{1}{2} \\left( f(q^-_\\imh) + f(q^+_\\imh) - \\frac{\\Dx}{\\Dt} (q^+_\\imh - q^-_\\imh)\\right)$$\n",
     "\n",
     "This provides second-order accuracy in space.  We also need to make the method second-order accurate in time.  We can do so by using a second-order Runge--Kutta method.  Then the full method is\n",
     "\n",
diff --git a/Lesson_04_Fluid_dynamics.ipynb b/Lesson_04_Fluid_dynamics.ipynb
index 2216085..90b9249 100644
--- a/Lesson_04_Fluid_dynamics.ipynb
+++ b/Lesson_04_Fluid_dynamics.ipynb
@@ -227,7 +227,7 @@
       "\n",
       "Like the momentum flux, the energy flux involves both bulk transport ($Eu$) and transport due to pressure ($pu$):\n",
       "\n",
-      "$$E_t + (u(E+p)) = 0.$$"
+      "$$E_t + (u(E+p))_x = 0.$$"
      ]
     },
     {