===============
Routing Methods
===============

SWIFlow has two routing methods developed in it.

* Direct
* Muskingum-Cunge (1969)

Direct
======

Direct routing assumes that a bulk of the time water is spent on the hill
slope. This is only valid for small catchments.

Muskingum-Cunge (Under Development)
===================================

Description of technique first provided by `Cunge 1969`_.
Numerical discretization taken from `Gallice et. al 2016`_ and re-transcribed
below.

*Variables*

* n - Time index [unitless]
* i - Discretize stream reach index [unitless]
* t - Time [Seconds]
* Q - Flowrate [M^3/S]
* l_i - Stream reach discretized length for stream i [m]
* w - Stream segment width [m]
* S_0 - Local bed slope []
* C_r - Wave celerity [m/s]
* Q_r - Representative discharge [m^3/s]
* n_m - Mannings coefficient ( 0.03 - 0.10 for small natural streams) [s/m^-1/3]


**Assumptions**

* C_r is derived from Q_r assuming rectangular channel cross section

.. math::

    Q^{n+1}_{i} = C_1 Q^{n}_{i-1} + C_2 Q^{n+1}_{i-1} + C_3 Q^{n}_{i}

Where

.. math::

    C_1 = \frac{k_i x_i + 0.5 \Delta t}{k_i (1-x_i) + 0.5\Delta t}

    C_2 = \frac{-k_i x_i + 0.5 \Delta t}{k_i (1-x_i) + 0.5\Delta t}

    C_3 = \frac{k_i (1-x_i) - 0.5 \Delta t}{k_i (1-x_i) + 0.5\Delta t}

Where

.. math::

    k_i = \frac{l_i}{c_r}

    c_r = \frac{5}{3} \bigg(\frac{S_0}{n_{m}^2}\bigg)^{\frac{3}{10}} \bigg(\frac{Q_r}{w}\bigg)^{\frac{2}{5}}

    x_i = 0.5 * min\bigg(1, 1 - \frac{Q_r}{c_r w S_0 l_i}\bigg)

    Q_r = \frac{Q^{n}_{i-1} + Q^{n+1}_{i-1} + Q^{n}_{i}}{3}

    h^{n+1}_i = \bigg(\frac{n_m Q^{n+1}_i}{w S_0}\bigg)^{\frac{3}{5}}

Numerical stability guidance provided by the following for lumped systems as
opposed to gridded discretized reaches:

.. math::

    2k_i x_i <= \Delta t <= 2k_i(1-x_i)

.. _Cunge 1969: https://www.tandfonline.com/doi/pdf/10.1080/00221686909500264
.. _Gallice et. al 2016: https://www.geosci-model-dev.net/9/4491/2016/gmd-9-4491-2016.pdf