add inviscid burger solver (#144)

add inviscid burger problem based on advection problem
the interface state prediction follows the incompressible solver closely

Author: zhichen3

README.md

@@ -132,6 +132,8 @@ pyro provides the following solvers (all in 2-d):
   - `advection_weno`: a method-of-lines WENO solver for linear
     advection.
 
+  - `burgers': a second-order unsplit solver for invsicid Burgers' equation.
+
   - `compressible`: a second-order unsplit solver for the Euler
     equations of compressible hydrodynamics.  This uses characteristic
 	tracing and corner coupling for the prediction of the interface

docs/source/burgers.png

@@ -0,0 +1,36 @@
+Burgers' Equation
+==================
+
+Burgers' Equation is a nonlinear hyperbolic equation. It has the same form as the advection equation, except that the quantity being advected is the velocity itself.
+
+``Inviscid Burgers``
+--------------------------------
+
+A 2D Burgers' Equation has the following form:
+
+.. math::
+
+   u_t + u u_x + v u_y = 0\\
+   v_t + u v_x + v v_y = 0
+
+Here we have two 2D advection equations, where the x-velocity, :math:`u`, and y-velocity, :math:`v`, are the two quantities that we wish to advect with.
+
+:py:mod:`pyro.burgers` is modified based on the :py:mod:`pyro.advection` with a different Riemann solver and timestep restriction.
+
+Since velocity is no longer a constant, the timestep is now restricted to the each minimum velocity in each cell:
+
+.. math::
+
+   \Delta t < \min \left \{ \min \left \{ \frac{\Delta x}{|u_i|} \right \}, \min \left \{ \frac{\Delta y}{|v_i|}  \right \}  \right \}
+
+The main difference of Burgers equation compared to the linear advection equation is the creation of shock and rarefactions due velocity been non-constant. This introduces a slightly different Riemann's problem which depends on shock speed by using the *Rankine-Hugoniot* jump condition.
+
+The parameters for this solver are:
+
+.. include:: burgers_defaults.inc
+
+
+.. image:: burgers.png
+   :align: center
+
+The figure above is generated using ``burgers/problems/test.py``, which is used to test the validity of the solver. Bottom-left of the domain has a higher velocity than the top-right domain. With :math:`u_{i,j}=v_{i,j}`, the wave travels diagonally to the top-right with a constant velocity that is equal to the shock speed. ``burgers/problem/verify.py`` can be used to calculate the wave speed using outputs from ``test.py`` and compare to the theoretical shock speed.

docs/source/burgers_basics.rst

@@ -0,0 +1,26 @@
+* section: [advection]
+
+  +----------------------------------+----------------+----------------------------------------------------+
+  | option                           | value          | description                                        |
+  +==================================+================+====================================================+
+  | limiter                          | ``2``          | limiter (0 = none, 1 = 2nd order, 2 = 4th order)   |
+  +----------------------------------+----------------+----------------------------------------------------+
+
+* section: [driver]
+
+  +----------------------------------+----------------+----------------------------------------------------+
+  | option                           | value          | description                                        |
+  +==================================+================+====================================================+
+  | cfl                              | ``0.8``        | advective CFL number                               |
+  +----------------------------------+----------------+----------------------------------------------------+
+
+* section: [particles]
+
+  +----------------------------------+----------------+----------------------------------------------------+
+  | option                           | value          | description                                        |
+  +==================================+================+====================================================+
+  | do_particles                     | ``0``          |                                                    |
+  +----------------------------------+----------------+----------------------------------------------------+
+  | particle_generator               | ``grid``       |                                                    |
+  +----------------------------------+----------------+----------------------------------------------------+
+

docs/source/burgers_defaults.inc

@@ -30,6 +30,7 @@ pyro: a python hydro code
    :caption: Solvers
 
    advection_basics
+   burgers_basics
    compressible_basics
    compressible_compare
    multigrid_basics

docs/source/index.rst

@@ -0,0 +1,26 @@
+pyro.burgers.problems package
+===============================
+
+.. automodule:: pyro.burgers.problems
+   :members:
+   :undoc-members:
+   :show-inheritance:
+
+Submodules
+----------
+
+pyro.burgers.problems.smooth module
+-------------------------------------
+
+.. automodule:: pyro.burgers.problems.smooth
+   :members:
+   :undoc-members:
+   :show-inheritance:
+
+pyro.burgers.problems.tophat module
+-------------------------------------
+
+.. automodule:: pyro.burgers.problems.tophat
+   :members:
+   :undoc-members:
+   :show-inheritance:

docs/source/pyro.burgers.problems.rst

@@ -0,0 +1,34 @@
+pyro.burgers package
+======================
+
+.. automodule:: pyro.burgers
+   :members:
+   :undoc-members:
+   :show-inheritance:
+
+Subpackages
+-----------
+
+.. toctree::
+   :maxdepth: 4
+
+   pyro.burgers.problems
+
+Submodules
+----------
+
+pyro.burgers.advective\_fluxes module
+---------------------------------------
+
+.. automodule:: pyro.burgers.advective_fluxes
+   :members:
+   :undoc-members:
+   :show-inheritance:
+
+pyro.burgers.simulation module
+--------------------------------
+
+.. automodule:: pyro.burgers.simulation
+   :members:
+   :undoc-members:
+   :show-inheritance:

docs/source/pyro.burgers.rst

@@ -17,6 +17,7 @@ Subpackages
    pyro.advection_nonuniform
    pyro.advection_rk
    pyro.advection_weno
+   pyro.burgers
    pyro.compressible
    pyro.compressible_fv4
    pyro.compressible_react

docs/source/pyro.rst

@@ -0,0 +1,7 @@
+"""
+The pyro inviscid burgers solver.  This implements a second-order,
+unsplit method for inviscid burgers equations based on the Colella 1990 paper.
+"""
+
+__all__ = ['simulation']
+from .simulation import Simulation

pyro/burgers/__init__.py

@@ -0,0 +1,11 @@
+[driver]
+cfl       = 0.8           ; advective CFL number
+
+
+[advection]
+limiter = 2  ; limiter (0 = none, 1 = 2nd order, 2 = 4th order)
+
+
+[particles]
+do_particles = 0
+particle_generator = grid