ader 1.2.1
The ADER method for solving any (potentially very stiff) hyperbolic system of PDEs
License: MIT
Keywords: ADER,WENO,Discontinuous Galerkin,Finite Volume,PDEs,Partial Differential Equations

ADER
====

A Python implementation of the ADER method for solving any (potentially very
stiff) hyperbolic system of PDEs of the following form:

.. figure:: https://github.com/haranjackson/ADER/raw/master/images/eq1.png
   :alt: Figure 1

An arbitrary number of spatial domains can be used, and this implementation is
capable of solving the equations to any order of accuracy. Second-order
parabolic PDEs will be implemented soon.

.. image:: https://travis-ci.org/haranjackson/ADER.svg?master
   :target: https://travis-ci.org/haranjackson/ADER
   :alt: Build Status

Installation
------------

Requires Python 3.6. Run ``pip install ader``

The following dependencies will be installed:

-  NumPy 1.14.5
-  SciPy 1.1.0
-  Tangent 0.1.9

Background
----------

Given cell-wise constant initial data defined on a computational grid, this
program performs an arbitrary-order polynomial reconstruction of the data in
each cell, according to a modified version of the WENO method, as presented in
[1].

To the same order, a spatio-temporal polynomial reconstruction of the data is
then obtained in each spacetime cell by the Discontinuous Galerkin method,
using the WENO reconstruction as initial data at the start of the timestep
(see [2,3]).

Finally, a finite volume update step is taken, using the DG reconstruction to
calculate the values of the intercell fluxes and non-conservative intercell
jump terms, and the interior source terms and non-conservative terms (see [3]).

The intercell fluxes and non-conservative jumps are calculated using either a
Rusanov-type flux [4], a Roe-type flux [5], or an Osher-type flux [6].

Usage
-----

Defining the System of Equations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The user must define the flux function (returning a NumPy array):

``F(Q, d, model_params)``

Given a vector of conserved variables ``Q``, ``F`` must return the flux in
direction ``d`` (where ``d=0,1,...`` corresponds to the x-,y-,... axes).
``model_params`` should be an object containing any other parameters required
by the model in the calculation of the flux (for example, the heat capacity
ratio in the Euler equations, or the viscosity in the Navier-Stokes equations).
``model_params`` does not have to be used, but it must be contained in the
signature of ``F``.

Similarly, if required, the nonconservative matrix B(Q) in direction d must be
defined as a square NumPy array, thus:

``B(Q, d, model_params)``

If required, the source terms must be defined as a NumPy array thus:

``S(Q, model_params)``

Note that the governing system of PDEs can be written as:

.. figure:: https://github.com/haranjackson/ADER/raw/master/images/eq2.png
   :alt: Figure 2

where the system Jacobian M corresponds to:

.. figure:: https://github.com/haranjackson/ADER/raw/master/images/eq3.png
   :alt: Figure 3

If an analytical form for M is known, it should be defined thus:

``M(Q, d, model_params)``

If an analytical form for the eigenvalue of M with the largest absolute value
is known, it should be defined thus:

``max_eig(Q, d, model_params)``

The system Jacobian and largest absolute eigenvalue are required for the ADER
method. If no analytical forms are available, they will be derived from the
supplied flux (and nonconservative matrix, if supplied) using `automatic
differentiation <https://en.wikipedia.org/wiki/Automatic_differentiation>`__.
This will introduce some computational overhead, however, and some
considerations need to be made when defining the flux function (see the
dedicated section below).

Solving the Equations
~~~~~~~~~~~~~~~~~~~~~

Suppose we are solving the `reactive Euler equations <https://www.sciencedirect.com/science/article/pii/0895717796001471>`__
in 2D. We define ``F`` and ``S`` as above (but not ``B``, as the equations are
conservative). We also define ``model_params`` to hold the heat capacity ratio
and reaction constants. These equations contain 5 conserved variables. To solve
them to order 3, with 4 CPU cores, we set up the solver object thus:

::

    from ader import Solver

    solver = Solver(nvar=5, ndim=2, F=F, B=None, S=S, model_params=model_params, order=3, ncore=4)

Analytical forms of the system Jacobian and the eigenvalue of largest absolute
size exist for the reactive Euler equations. If we define these in ``M`` and
``max_eig`` we may instead create the solver object thus:

::

    solver = Solver(nvar=5, ndim=2, F=F, B=None, S=S, M=M, max_eig=max_eig, model_params=model_params, order=3, ncore=4)

To solve a particular problem, we must define the initial state,
``initial_grid``. This must be a NumPy array with 3 axes, such that
``initial_grid[i,j,k]`` is equal to the value of the kth conserved variable in
cell (i,j). We must also define list ``dX=[dx,dy]`` where dx,dy are the grid
spacing in the x and y axes. To solve the problem to a final time of 0.5, with
a CFL number of 0.9, while printing all output, we call:

::

    solution = solver.solve(initial_grid, 0.5, dX, cfl=0.9, verbose=True)

Advanced Usage
~~~~~~~~~~~~~~

The Solver class has the following additional arguments:

-  **riemann\_solver** (default 'rusanov'): Which Riemann solver should be
   used. Options: 'rusanov', 'roe', 'osher'.
-  **stiff\_dg** (default False): Whether to use a Newton Method to solve the
   root finding involved in calculating the DG predictor.
-  **stiff\_dg\_guess** (default False): Whether to use an advanced initial
   guess for the DG predictor (only for very stiff systems).
-  **newton\_dg\_guess** (default False): Whether to compute the advanced
   initial guess using a Newton Method (only for very, very stiff systems).
-  **DG\_TOL** (default 6e-6): The tolerance to which the DG predictor is
   calculated.
-  **DG\_MAX\_ITER** (default 50): Maximum number of iterations attempted if
   solving the DG root finding problem iteratively (not with a Newton Method)
-  **WENO\_r** (default 8): The WENO exponent r.
-  **WENO\_λc** (default 1e5): The WENO weighting of the central stencils.
-  **WENO\_λs** (default 1): The WENO weighting of the side stencils.
-  **WENO\_ε** (default 1e-14): The constant used in the WENO method to avoid
   numerical issues.

The Solver.solve method has the following additional arguments:

-  **boundary\_conditions** (default 'transitive'): Which kind of boundary
   conditions to use. Options: 'transitive', 'periodic',
   ``func(grid, N, ndim)``. In the latter case, the user defines a function
   with the stated signature. It should return a NumPy array with the same
   number of axes as grid, but with ``N`` more cells on either side of the grid
   in each spatial direction, where ``N`` is equal to the order of the method
   being used. These extra cells are required by an N-order method.
-  **callback** (default None): A user-defined callback function with signature
   ``callback(grid, t, count)`` where ``grid`` is the value of the
   computational grid at time ``t`` (and timestep ``count``).

Examples
~~~~~~~~

Check out example.py to see a couple of problems being solved for the GPR model
and the reaction Euler equations.

Notes
-----

Speed
~~~~~

This implementation is pretty slow. It's really only intended to be used only
for academic purposes. If you have a commercial application that requires a
rapid, bullet-proof implementation of the ADER method or the GPR model, then
get in touch (jackson.haran@gmail.com).

Automatic Differentiation
~~~~~~~~~~~~~~~~~~~~~~~~~

The automatic differentiation used to derive ``M`` and ``max_eig`` is
performed using `Google's Tangent library <https://github.com/google/tangent>`__.
Although it's great, this library is quite new, and it cannot cope with all
operations that you may use in your fluxes (although development is proceeding
quickly). In particular, it will never be able to handle closures, and classes
are not yet implemented. Some NumPy functions such as ``inv`` have not yet been
implemented. If you run into issues, drop me a quick message and I'll let you
know if I can make it work.

References
----------

1. Dumbser, Zanotti, Hidalgo, Balsara - *ADER-WENO finite volume schemes with
   space-time adaptive mesh refinement*
2. Dumbser, Castro, Pares, Toro - *ADER schemes on unstructured meshes for
   nonconservative hyperbolic systems: Applications to geophysical flows*
3. Dumbser, Hidalgo, Zanotti - *High order space-time adaptive ADER-WENO finite
   volume schemes for non-conservative hyperbolic systems*
4. Toro - *Riemann Solvers and Numerical Methods for Fluid Dynamics: A
   Practical Introduction*
5. Dumbser, Toro - *On Universal Osher-Type Schemes for General Nonlinear
   Hyperbolic Conservation Laws*
6. Dumbser, Toro - *A simple extension of the Osher Riemann solver to
   non-conservative hyperbolic systems*