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‎source/3_numerical.rst‎

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@@ -74,7 +74,7 @@ This yields a global error of order :math:`O(\Delta t^4)` with four velocity eva
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**Sixth-Order Runge-Kutta (RK6)**
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The six-stage scheme uses non-uniform weights to attain sixth-order accuracy:
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The seven-stage scheme uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy. This method originates from
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.. math::
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@@ -86,34 +86,27 @@ The six-stage scheme uses non-uniform weights to attain sixth-order accuracy:
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k_6 = \mathbf{u}\!\bigl(\mathbf{x}_n + \Delta t(-\tfrac{3}{2}k_1 + 2k_2 - \tfrac{1}{2}k_3 + k_4),\;t_n + \Delta t\bigr),\\
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\mathbf{x}_{n+1} = \mathbf{x}_n + \Delta t\bigl(\tfrac{1}{20}k_1 + \tfrac{1}{4}k_4 + \tfrac{1}{5}k_5 + \tfrac{1}{2}k_6\bigr).
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This scheme incurs a global error of order :math:`O(\Delta t^6)` with six velocity evaluations.
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``output_dir`` | Path to directory for saving results | ``<string>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``auto_wall`` | Flag for automatic wall detection | ``0`` | Manually indicate computation bounds. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| | | (``1``) | Auto detect bounds from input mesh. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``wall_xrange`` | X-direction wall bounds [xmin, xmax] | ``[<float>, <float>]`` | Dimensionalized coordinates bounds. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``wall_yrange`` | Y-direction wall bounds [ymin, ymax] | ``[<float>, <float>]`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``wall_zrange`` | Z-direction wall bounds [zmin, zmax] | ``[<float>, <float>]`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``x`` | Number of Lagrangian seed points in X direction | ``<int>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``y`` | Number of Lagrangian seed points in Y direction | ``<int>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``z`` | Number of Lagrangian seed points in Z direction | ``<int>`` | |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``T_range`` | Time range [``T_start``, ``T_end``] in frames | ``[<float>, <float>]`` | ``T_end`` no larger than ``N_frames``. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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| ``dt`` | Time step for particle advection in frames | ``<float>`` | |
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+===================+======================================================+=========================+=========================================================================================================+
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| ``physical_dt`` | Physical time step between two input frames | ``<float>`` | Perform physical advection. |
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+-------------------+------------------------------------------------------+-------------------------+---------------------------------------------------------------------------------------------------------+
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This scheme incurs a global error of order with six velocity evaluations.
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`0` | | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/10` | | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`\mp\sqrt{5}/10` | :math:`(5\pm2\sqrt{5})/10` | | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(-15\pm7\sqrt{5})/20` | :math:`(-1\pm\sqrt{5})/4` | :math:`(15\mp7\sqrt{5})/10` | | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\pm\sqrt{5})/10` | :math:`(5\mp\sqrt{5})/60` | :math:`0` | :math:`1/6` | :math:`(15\pm7\sqrt{5})/60` | | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`(5\mp\sqrt{5})/10` | :math:`(5\pm\sqrt{5})/60` | :math:`0` | :math:`(9\mp5\sqrt{5})/12` | :math:`1/6` | :math:`(-5\pm3\sqrt{5})/10` | |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`1` | :math:`1/6` | :math:`0` | :math:`(-55\pm25\sqrt{5})/12` | :math:`(-25\mp7\sqrt{5})/12` | :math:`5\mp2\sqrt{5}` | :math:`(5\pm\sqrt{5})/2` |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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| :math:`1/12` | :math:`0` | :math:`0` | :math:`0` | :math:`5/12` | :math:`5/12` | :math:`1/12` |
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+
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.. _ftlefinal:

‎source/9_references.rst‎

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.. [Lekien2005] Lekien, F., Marsden, J. "Tricubic interpolation in three dimensions," *International Journal for Numerical Methods in Engineering* **63** (2005).
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.. [Butcher] Butcher J. C. "On Runge-Kutta processes of high order." *Journal of the Australian Mathematical Society* **4(2)** (1964). doi:10.1017/S1446788700023387
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Papers Who Uses ``Py3DFTLE``

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