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‎source/3_numerical.rst‎

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@@ -99,7 +99,7 @@ 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 seven-stage scheme ``ERK6(7)`` uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy. This method originates from [Butcher]_.
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The seven-stage scheme ``ERK6(7)`` uses non-uniform weights to attain global :math:`O(\Delta t^6)` accuracy, originating from [Butcher1964]_.
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As for the coefficients for ``RK6`` are more complex to write into equations, the Butcher table is given as follows.
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+-----------------------------+-------------------------------+----------------------------+---------------------------------+------------------------------+-----------------------------+--------------------------+-------------------------+
@@ -122,8 +122,27 @@ As for the coefficients for ``RK6`` are more complex to write into equations, th
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| :math:`b_i` | :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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In our computation, the up symbol side is applied, in other words, ``±`` represents ``+``, taking :math:`\lambda=+\sqrt{5}`.
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In our computation, the up symbol side is applied, in other words, ``±`` represents ``+``, taking :math:`\lambda=+\sqrt{5}`. With 15 digis are kept, the Butcher table used by the author is shown in the following table.
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| :math:`c_i` | :math:`a_{i1}` | :math:`a_{i2}` | :math:`a_{i3}` | :math:`a_{i4}` | :math:`a_{i5}` | :math:`a_{i6}` | :math:`a_{i7}` |
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+===============================+===============================+===============================+===============================+===============================+===============================+===============================+===============================+
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| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.276393202250021 | 0.276393202250021 | 0 | 0 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.723606797749979 | -0.223606797749979 | 0.947213595499958 | 0 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.276393202250021 | 0.0326237921249264 | 0.309016994374947 | -0.0652475842498529 | 0 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.723606797749979 | 0.0460655337083368 | 0 | 0.166666666666667 | 0.510874597374975 | 0 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 0.276393202250021 | 0.12060113295833 | 0 | -0.181694990624912 | 0.166666666666667 | 0.170820393249937 | 0 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| 1 | 0.166666666666667 | 0 | 0.0751416197912285 | -3.38770632020821 | 0.52786404500042 | 3.61803398874989 | 0 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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| b_i | 0.0833333333333333 | 0 | 0 | 0 | 0.416666666666667 | 0.416666666666667 | 0.0833333333333333 |
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+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+-------------------------------+
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‎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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.. [Butcher1964] 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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