Changes between Version 7 and Version 8 of doc/tec/rk3

Timestamp:

Jan 10, 2011 2:02:18 PM (14 years ago)

Author:

suehring

Comment:

--

doc/tec/rk3

@@ -34 +34 @@
 \[ \hat\psi_{2} & = & \hat\psi_{1} + \frac{1}{48} \Delta t \left( 45 f\left(\hat\psi_1\right) - 25 f\left(\psi_{n}\right) \right) \]
 \[ f\left(\hat\psi_{1}\right) & = &-153 f\left(\hat\psi_{1}\right) + 85 f\left(\psi_{n}\right) \]
-\[ \hat\psi_{3} & = & \left( \psi_{n+1} \right) = \hat\psi_{2} + \frac{1}{240} \Delta t \left( 128 f\left(\hat\psi_2\right) + 16 f \left(\hat\psi_{1}\right) \right) \]
+\[ \hat\psi_{3} & = & \left( \psi_{n+1} \right) = \hat\psi_{2} + \frac{1}{240} \Delta t \left( 128 f\left(\hat\psi_2\right) + 15 f \left(\hat\psi_{1}\right) \right) \]
 }}}
 
-For reasons of clarity the time integration for several schemes ( further scheme are: Leap-frog, Euler and 2^nd^ order Runge-Kutta scheme ) is implemented as following ( here for example for the u-component of velocity ):
+For reasons of clarity the time integration for several schemes (further schemes are: Leap-frog, Euler and 2^nd^ order Runge-Kutta scheme) is implemented as following (here for example the u-component of velocity):
 
 {{{
 #!Latex
 \begin{split}
- u_p\left(k,j,i\right) = \left(1.0 - tsc\left(1\right) \right) * u\_m\left(k,j,i\right) + tsc\left(1\right) * u\left(k,j,i\right) + dt\_3d * \left( \\ 
-                             tsc\left(2\right) * tend\left(k,j,i\right) + tsc\left(3\right) * tu\_m\left(k,j,i\right)  \\
-                             + tsc\left(4\right) * \left(p\left(k,j,i)\right) - p\left(k,j,i-1\right) \right) * ddx \ \ ) \\
-                             - tsc\left(5\right) * rdf\left(k\right) * \left(u\left(k,j,i) - ug \right) 
+ \textnormal{u}\_\textnormal{p}\left(k,j,i\right) = \left(1.0 - \textnormal{tsc}\left(1\right) \right) * \textnormal{u}\_\textnormal{m}\left(k,j,i\right) + \textnormal{tsc}\left(1\right) * \textnormal{u}\left(k,j,i\right) + \textnormal{dt}\_\textnormal{3d}* \left( \\ 
+                             \textnormal{tsc}\left(2\right) * \textnormal{tend}\left(k,j,i\right) + \textnormal{tsc}\left(3\right) * \textnormal{tu}\_\textnormal{m}\left(k,j,i\right)  \\
+                             + \textnormal{tsc}\left(4\right) * \left(\textnormal{p}\left(k,j,i)\right) - \textnormal{p}\left(k,j,i-1\right) \right) * ddx \ \ ) \\
+                             - \textnormal{tsc}\left(5\right) * \textnormal{rdf}\left(k\right) * \left(\textnormal{u}\left(k,j,i) - \textnormal{ug} \right) 
 \end{split}
 }}}
 
-and steered by the array tsc
+and steered by the array tsc(1:5)
+
+|| tsc(1) || tsc(2) || tsc(3) || tsc(4) || tsc(5) ||
+|| 1 || 1/3 || 0 || 0 || 0 || 1^st^ substep
+|| 1 || 15/16 || -25/48 || 0 || 0 || 2^nd^ substep
+|| 1 || 8/15 || 1/15 || 0 || 1 || 3^rd^ substep