Merged from upstream

Merge remote-tracking branch 'upstream/master'

Euler_approximate_solvers.ipynb

@@ -21,7 +21,7 @@
     "editable": true
    },
    "source": [
-    "In the [last notebook](Euler_equations.ipynb) we studied the Euler equations of inviscid, compressible fluid flow -- including the Riemann problem and its solution.  As we saw, the exact solution of the Riemann problem is computationally expensive, since it requires solving a set of nonlinear algebraic equations.  In the context of finite volume methods, the detailed structure of the Riemann solution is almost immediately discarded -- only its impact on the neighboring cell averages is used.  So it makes sense to consider whether we can approximate the solution with less computation.  In this notebook, we investigate approximate solvers for the Euler equations."
+    "In the [Part I](Euler_equations.ipynb) we studied the Riemann problem for Euler equations of inviscid, compressible fluid flow .  As we saw, the exact solution of the Riemann problem is computationally expensive, since it requires solving a set of nonlinear algebraic equations.  In the context of finite volume methods, the detailed structure of the Riemann solution is almost immediately discarded -- only its impact on the neighboring cell averages is used.  So it makes sense to consider whether we can approximate the solution with less computation.  In this chapter, we investigate approximate solvers for the Euler equations."
    ]
   },
   {

Index.ipynb

@@ -23,6 +23,8 @@
     "  \n",
     "## Part II: Approximate solvers\n",
     "\n",
+    "  1. [Euler approximate solvers](Euler_approximate_solvers.ipynb)\n",
+    "\n",
     "## Part III: Riemann problems with spatially-varying flux\n",
     "  1. Advection\n",
     "  2. Acoustics\n",
@@ -35,10 +37,12 @@
     " \n",
     "## Part V: Non-classical problems\n",
     "  1. [Nonconvex flux for a scalar problem](Nonconvex_Scalar_Osher_Solution.ipynb)\n",
+    "  2. [Pressureless flow](Pressureless_flow.ipynb)\n",
     " \n",
     "## Part VI: Multidimensional systems\n",
     "  1. [Acoustics](http://nbviewer.jupyter.org/github/maojrs/ipynotebooks/blob/master/acoustics_riemann.ipynb)\n",
-    "  2. [Elasticity](http://nbviewer.jupyter.org/github/maojrs/ipynotebooks/blob/master/elasticity_riemann.ipynb)"
+    "  2. [Elasticity](http://nbviewer.jupyter.org/github/maojrs/ipynotebooks/blob/master/elasticity_riemann.ipynb)\n",
+    "  3. [The Kitchen Sink: shallow water in cylindrical coordinates](Kitchen_sink_problem.ipynb)"
    ]
   }
  ],

Kitchen_sink_problem.ipynb

@@ -19,7 +19,7 @@
     "from clawpack import riemann\n",
     "from clawpack.visclaw.ianimate import ianimate\n",
     "import matplotlib\n",
-    "matplotlib.rcParams['font.size'] = 12\n",
+    "plt.style.use('seaborn-talk')\n",
     "from IPython.display import HTML"
    ]
   },
@@ -745,17 +745,6 @@
     "plt.xlim(r[0],r[-1]);"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "collapsed": false,
-    "deletable": true,
-    "editable": true
-   },
-   "outputs": [],
-   "source": []
-  },
   {
    "cell_type": "markdown",
    "metadata": {

README.md

@@ -17,9 +17,9 @@ Parentheticals indicate concepts introduced for the first time.
 1. [Acoustics](https://github.com/clawpack/riemann_book/blob/master/Acoustics.ipynb) (eigenvalue analysis, characteristics, similarity solutions)
 2. [Traffic flow](https://github.com/clawpack/riemann_book/blob/master/Traffic_flow.ipynb) (shocks, rarefactions, conservation, jump conditions)
 3. Burgers' (weak solutions)
-4. [Shallow water](https://github.com/clawpack/riemann_book/blob/master/Shallow_tracer.ipynb) (jump conditions for a nonlinear system; Riemann invariants, integral curves, Hugoniot Loci) (see also [this](http://nbviewer.jupyter.org/url/faculty.washington.edu/rjl/notebooks/shallow/SW_riemann_tester.ipynb) and [this](http://nbviewer.jupyter.org/gist/rjleveque/8994740))
+4. [Shallow water](https://github.com/clawpack/riemann_book/blob/master/Shallow_water.ipynb) (jump conditions for a nonlinear system; Riemann invariants, integral curves, Hugoniot Loci) (see also [this](http://nbviewer.jupyter.org/url/faculty.washington.edu/rjl/notebooks/shallow/SW_riemann_tester.ipynb) and [this](http://nbviewer.jupyter.org/gist/rjleveque/8994740))
 5. How to solve the Riemann problem exactly -- go in depth into SW solver, including Newton iteration to find root of piecewise function
-5. Shallow water with a tracer (contact waves)
+5. [Shallow water with a tracer](https://github.com/clawpack/riemann_book/blob/master/Shallow_tracer.ipynb) (contact waves)
 5. [Euler equations](https://github.com/clawpack/riemann_book/blob/master/Euler_equations.ipynb)
 
 **Part II: Approximate solvers**

Shallow_water.ipynb

@@ -532,7 +532,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Comparison of integral curves and Hugoniot loci\n",
     "You may have noticed that the integral curves look very similar to the Hugoniot loci, especially when plotted in the $h-hu$ plane.  Below we plot the curve from each of these families that passes through a specified point.  This divergence is less noticeable in the $h-hu$ plane since all of the curves approach the origin."
@@ -542,7 +545,9 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -561,15 +566,18 @@
     "    plt.show()\n",
     "    \n",
     "interact(compare_curves,\n",
-    "         wave_family=widgets.RadioButtons(options=[1,2]),\n",
+    "         wave_family=widgets.RadioButtons(options=[1,2],description='Wave family:'),\n",
     "         y_axis=widgets.RadioButtons(options=['u','hu'],description='Vertical axis:'),\n",
     "         h0=widgets.FloatSlider(min=1.e-1,max=3.,value=1.,description='$h_*$'),\n",
     "         u0=widgets.FloatSlider(min=-3,max=3,description='$u_*$'));"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Near the point of intersection, the curves are very close; indeed, they must be tangent at this point since their direction is parallel to the corresponding eigenvector there.  Far from this point they diverge; for small depths they must diverge greatly, since the Hugoniot locus never reaches $h=0$ at any finite depth."
    ]
@@ -706,7 +714,7 @@
     "editable": true
    },
    "source": [
-    "Plot the 2-characteristics and notice that they cross each other within the 2-rarefaction.  This rarefaction is not physical and should be replaced with a shock; the corresponding part of the integral curve is hence shown as a dashed line."
+    "Plot the 2-characteristics and notice that they cross each other within the 2-rarefaction. This means that the solution we constructed is triple-valued and nonsensical as a solution to this one-dimensional conservation law, and so this portion of the solution is omitted in the plots of depth and momentum.  In this case a rarefaction wave is not physical and should be replaced with a shock; the corresponding part of the integral curve is hence shown as a dashed line."
    ]
   },
   {
@@ -792,7 +800,7 @@
     "    shallow_water.phase_plane_curves(q_r[0], q_r[1], 'qright', wave_family=2,ax=ax[0])\n",
     "    shallow_water.phase_plane_curves(q_l[0], q_l[1], 'qleft', wave_family=1,y_axis='hu',ax=ax[1])\n",
     "    shallow_water.phase_plane_curves(q_r[0], q_r[1], 'qright', wave_family=2,y_axis='hu',ax=ax[1])\n",
-    "    ax[0].set_title('h-u plane'); ax[1].set_title('h-hu plane')\n",
+    "    ax[0].set_title('h-u plane'); ax[1].set_title('h-hu plane'); ax[0].set_xlim(0,3); ax[1].set_xlim(0,3);\n",
     "    ax[0].set_ylim(-10,10); ax[1].set_ylim(-10,10); plt.tight_layout(); plt.show()\n",
     "interact(connect_states,\n",
     "         h_l=widgets.FloatSlider(min=0.001,max=2,value=1),\n",
@@ -810,6 +818,8 @@
    "source": [
     "You should find that by making the initial states flow sufficiently fast away from each other, there is no intersection in the $h-u$ plane.  In the $h-hu$ plane, the integral curves always intersect at the origin.  The reason for this ambiguity is that, for zero depth it isn't meaningful to assign a velocity.  Thus in the $h-u$ plane we could think of the entire $u$-axis as being part of every integral curve.  That means that we can always connect the left and right states via an intermediate state with depth $h=0$ (a dry state).  \n",
     "\n",
+    "Since the 1-integral curve through $q_l$ reaches $h=0$ at $u = u_l + 2 \\sqrt{gh_l}$ and the 2-integral curve reaches $h=0$ at $u=u_r-2\\sqrt{gh_r}$, we see that a dry middle state occurs if and only if $u_l + 2 \\sqrt{gh_l} < u_r-2\\sqrt{gh_r}$.\n",
+    "\n",
     "It is clear in this case that the solution must consist of two centered rarefaction waves, connecting the left and right states to the middle dry state.  These centered rarefactions still have the structure derived above; to complete the solution we only need to know the range of centered characteristics included in each rarefaction.  This is most conveniently determined by considering the $h-u$ plane.  The 1-integral curve passing through $q_l$ is given by $u = u_l + 2(\\sqrt{gh_l}-\\sqrt{gh_r})$; it reaches a dry state $h=0$ at $u=u_l + 2\\sqrt{gh_l}$.  Similarly, the 2-integral curve passing through $q_r$ reaches a dry state $h=0$ at $u=u_r - 2\\sqrt{gh_r}$.  Thus the dry state solution is given by\n",
     "\n",
     "\\begin{align}\n",
@@ -944,7 +954,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython2",
-   "version": "2.7.13"
+   "version": "2.7.12"
   }
  },
  "nbformat": 4,

exact_solvers/shallow_water.py

@@ -13,13 +13,14 @@ def primitive_to_conservative(h, u):
     hu = h*u
     return h, hu
 
-
 def conservative_to_primitive(h, hu):
     assert np.all(h>=0)
     # We should instead check that hu is zero everywhere that h is
     u = hu/pospart(h)
     return h, u
 
+def cons_to_prim(q):
+    return conservative_to_primitive(*q)
 
 def exact_riemann_solution(q_l, q_r, grav=1., force_waves=None, primitive_inputs=False):
     """Return the exact solution to the Riemann problem with initial states q_l, q_r.
@@ -173,18 +174,29 @@ def raref2(xi):
         speeds[1] = (ws[2],ws[3])
 
     def reval(xi):
+        """
+        Function that evaluates the Riemann solution for arbitrary xi = x/t.
+        Sets the solution to nan in an over-turning rarefaction wave
+        for illustration purposes of this non-physical solution.
+        """
         rar1 = raref1(xi)
         rar2 = raref2(xi)
         h_out = (xi<=ws[0])*h_l + \
             (xi>ws[0])*(xi<=ws[1])*rar1[0] + \
+            (xi>ws[1])*(xi<=ws[0])*1e9 +  \
             (xi>ws[1])*(xi<=ws[2])*h_m +  \
             (xi>ws[2])*(xi<=ws[3])*rar2[0] +  \
+            (xi>ws[3])*(xi<=ws[2])*1e9 +  \
             (xi>ws[3])*h_r
+        h_out[h_out>1e8] = np.nan
         hu_out = (xi<=ws[0])*hu_l + \
             (xi>ws[0])*(xi<=ws[1])*rar1[1] + \
+            (xi>ws[1])*(xi<=ws[0])*1e9 +  \
             (xi>ws[1])*(xi<=ws[2])*hu_m +  \
             (xi>ws[2])*(xi<=ws[3])*rar2[1] +  \
+            (xi>ws[3])*(xi<=ws[2])*1e9 +  \
             (xi>ws[3])*hu_r
+        hu_out[hu_out>1e8] = np.nan
         return h_out, hu_out
 
     return states, speeds, reval, wave_types
@@ -217,17 +229,18 @@ def hugoniot_locus(h, hstar, hustar, wave_family, g=1., y_axis='u'):
     d = np.sqrt(g*hstar*(1 + alpha/hstar)*(1 + alpha/(2*hstar)))
     if wave_family == 1:
         if y_axis == 'u':
-            return (hustar + alpha*(ustar - d))/h
+            return (hustar + alpha*(ustar - d))/pospart(h)
         else:
             return hustar + alpha*(ustar - d)
     else:
         if y_axis == 'u':
-            return (hustar + alpha*(ustar + d))/h
+            return (hustar + alpha*(ustar + d))/pospart(h)
         else:
             return hustar + alpha*(ustar + d)
 
 
-def phase_plane_curves(hstar, hustar, state, wave_family='both', y_axis='u', ax=None):
+def phase_plane_curves(hstar, hustar, state, g=1., wave_family='both', y_axis='u', ax=None,
+                       plot_unphysical=False):
     """
     Plot the curves of points in the h - u or h-hu phase plane that can be
     connected to (hstar,hustar).
@@ -241,37 +254,37 @@ def phase_plane_curves(hstar, hustar, state, wave_family='both', y_axis='u', ax=
     h = np.linspace(0, hstar, 200)
 
     if wave_family in [1,'both']:
-        if state == 'qleft':
-            u = integral_curve(h, hstar, hustar, 1, y_axis=y_axis)
+        if state == 'qleft' or plot_unphysical:
+            u = integral_curve(h, hstar, hustar, 1, g, y_axis=y_axis)
             ax.plot(h,u,'b', label='1-rarefactions')
-        else:
-            u = hugoniot_locus(h, hstar, hustar, 1, y_axis=y_axis)
+        if state == 'qright' or plot_unphysical:
+            u = hugoniot_locus(h, hstar, hustar, 1, g, y_axis=y_axis)
             ax.plot(h,u,'--r', label='1-shocks')
 
     if wave_family in [2,'both']:
-        if state == 'qleft':
-            u = hugoniot_locus(h, hstar, hustar, 2, y_axis=y_axis)
+        if state == 'qleft' or plot_unphysical:
+            u = hugoniot_locus(h, hstar, hustar, 2, g, y_axis=y_axis)
             ax.plot(h,u,'--r', label='2-shocks')
-        else:
-            u = integral_curve(h, hstar, hustar, 2, y_axis=y_axis)
+        if state == 'qright' or plot_unphysical:
+            u = integral_curve(h, hstar, hustar, 2, g, y_axis=y_axis)
             ax.plot(h,u,'b', label='2-rarefactions')
 
     h = np.linspace(hstar, 3, 200)
 
     if wave_family in [1,'both']:
-        if state == 'qright':
-            u = integral_curve(h, hstar, hustar, 1, y_axis=y_axis)
+        if state == 'qright' or plot_unphysical:
+            u = integral_curve(h, hstar, hustar, 1, g, y_axis=y_axis)
             ax.plot(h,u,'--b', label='1-rarefactions')
-        else:
-            u = hugoniot_locus(h, hstar, hustar, 1, y_axis=y_axis)
+        if state == 'qleft' or plot_unphysical:
+            u = hugoniot_locus(h, hstar, hustar, 1, g, y_axis=y_axis)
             ax.plot(h,u,'r', label='1-shocks')
 
     if wave_family in [2,'both']:
-        if state == 'qright':
-            u = hugoniot_locus(h, hstar, hustar, 2, y_axis=y_axis)
+        if state == 'qright' or plot_unphysical:
+            u = hugoniot_locus(h, hstar, hustar, 2, g, y_axis=y_axis)
             ax.plot(h,u,'r', label='2-shocks')
-        else:
-            u = integral_curve(h, hstar, hustar, 2, y_axis=y_axis)
+        if state == 'qleft' or plot_unphysical:
+            u = integral_curve(h, hstar, hustar, 2, g, y_axis=y_axis)
             ax.plot(h,u,'--b', label='2-rarefactions')
 
     # plot and label the point (hstar, hustar)
@@ -315,11 +328,11 @@ def phase_plane_plot(q_l, q_r, g=1., ax=None, force_waves=None, y_axis='u'):
                     (dry_velocity_l+dry_velocity_r)
 
     xmax, xmin = max(x), min(x)
-    ymax, ymin = max(y), min(y)
-    dx, dy = xmax - xmin, ymax - ymin
+    ymax = max(abs(y))
+    dx = xmax - xmin
     ymax = max(abs(y))
     ax.set_xlim(0, xmax + 0.5*dx)
-    ax.set_ylim(-ymax - 0.5*dy, ymax + 0.5*dy)
+    ax.set_ylim(-1.5*ymax, 1.5*ymax)
     ax.set_xlabel('Depth (h)')
     if y_axis == 'u':
         ax.set_ylabel('Velocity (u)')
@@ -359,7 +372,7 @@ def phase_plane_plot(q_l, q_r, g=1., ax=None, force_waves=None, y_axis='u'):
         ax.plot(xp,yp,'ok',markersize=10)
     # Label states
     for i,label in enumerate(('Left', 'Middle', 'Right')):
-        ax.text(x[i] + 0.025*dx,y[i] + 0.025*dy,label)
+        ax.text(x[i] + 0.025*dx,y[i] + 0.025*ymax,label)
 
 def plot_hugoniot_loci(plot_1=True,plot_2=False,y_axis='hu'):
     h = np.linspace(0.001,3,100)

utils/riemann_tools.py

@@ -288,8 +288,8 @@ def plot_riemann(states, s, riemann_eval, wave_types=None, t=0.1, ax=None,
 
     for i in range(num_vars):
         ax[i+1].set_xlim((-1,1))
-        qmax = q_sample[i][:].max()
-        qmin = q_sample[i][:].min()
+        qmax = max(np.nanmax(q_sample[i][:]), np.nanmax(states[i,:]))
+        qmin = min(np.nanmin(q_sample[i][:]), np.nanmin(states[i,:]))
         qdiff = qmax - qmin
         ax[i+1].set_xlim(-xmax,xmax)
         ax[i+1].set_ylim((qmin-0.1*qdiff,qmax+0.1*qdiff))
@@ -299,6 +299,19 @@ def plot_riemann(states, s, riemann_eval, wave_types=None, t=0.1, ax=None,
             ax[i+1].set_ylabel(variable_names[i])
 
     x = np.linspace(-xmax,xmax,1000)
+    # Make sure we have a point between each pair of waves
+    # Important e.g. for nearly-pressureless gas
+    wavespeeds = []
+    for speed in s:
+        from numbers import Number
+        if isinstance(speed, Number):
+            wavespeeds.append(speed)
+        else:
+            wavespeeds += speed
+    wavespeeds = np.array(wavespeeds)
+    xm = 0.5*(wavespeeds[1:]+wavespeeds[:-1])*t
+    iloc = np.searchsorted(x,xm)
+    x = np.insert(x, iloc, xm)
 
     if t == 0:
         q = riemann_eval(x/1e-10)
@@ -309,7 +322,10 @@ def plot_riemann(states, s, riemann_eval, wave_types=None, t=0.1, ax=None,
         q = derived_variables(q)
 
     for i in range(num_vars):
-        ax[i+1].plot(x,q[i][:],'-k',lw=2)
+        if color == 'multi':
+            ax[i+1].plot(x,q[i][:],'-k',lw=2)
+        else:
+            ax[i+1].plot(x,q[i][:],'-',color=color,lw=2)
         if i in fill:
             ax[i+1].fill_between(x,q[i][:],color='b')
             ax[i+1].set_ybound(lower=0)