Changes between Version 4 and Version 5 of doc/tec/rk3

Timestamp:

Jan 10, 2011 12:08:37 PM (14 years ago)

Author:

suehring

Comment:

--

doc/tec/rk3

@@ -28 +28 @@
 ||      ||  1/6  ||  3/10 || 8/15 ||
 
-For the implementation it is advantageous to compute ψ^N^ from the intermediate solutions ψ^1^ and ψ^2^ and combine the local tendencies in one array to save storage (therefor low-storage scheme) as follows: 
+For the implementation it is advantageous to compute ψ^N^ from the intermediate solutions ψ^1^ and ψ^2^ and combine the local tendencies in one array after the second substep to save storage (therefor low-storage scheme) as follows: 
 {{{
 #!Latex
 \[ \hat\psi_{1} & = & \psi_{n} + \frac{1}{3} \Delta t f\left(\psi_{n}\right) \]
-\[ \hat\psi_{2} & = & \hat\psi_{1} + \frac{1}{48} \Delta t \left( 45 f\left(\hat\psi_1\right) - 25 f\left(\psi_{n}\right) \]
-f\left(\hat\psi_{1}\right) & = &-153 f\left(\hat\psi_{1}\right) + 85 f\left(\psi_{n}\right) \]
-\[ \hat\psi_{3} & = & \left( \psi_{n+1} \right) = \hat\psi_{2} + \frac{1}{240} \Delta t \left( 128 f\left(\hat\psi_2\right) + 16 \left(\hat\psi_{1}\right) \]
+\[ \hat\psi_{2} & = & \hat\psi_{1} + \frac{1}{48} \Delta t \left( 45 f\left(\hat\psi_1\right) - 25 f\left(\psi_{n}\right) \right) \]
+\[ f\left(\hat\psi_{1}\right) & = &-153 f\left(\hat\psi_{1}\right) + 85 f\left(\psi_{n}\right) \]
+\[ \hat\psi_{3} & = & \left( \psi_{n+1} \right) = \hat\psi_{2} + \frac{1}{240} \Delta t \left( 128 f\left(\hat\psi_2\right) + 16 f \left(\hat\psi_{1}\right) \right) \]
 }}}
+
+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 ):
+
+{{{
+#!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 \right) \\
+                             - tsc\left(5\right) * rdf\left(k\right) * \left(u\left(k,j,i) - ug \right) 
+\end{split}
+}}}
+
+and steered by the array tsc
+