Changes between Version 13 and Version 14 of doc/tec/rk3

Timestamp:

Nov 12, 2015 2:26:40 PM (10 years ago)

Author:

boeske

Comment:

--

doc/tec/rk3

@@ -5 +5 @@
 {{{
 #!Latex
-\[  \frac{d \psi}{d t} & = & f(t,\psi) \]
+$\dfrac{d \psi}{d t} = f(t,\psi)$
 }}}
 as follows ( Baldauf, 2008 ):
 {{{
 #!Latex
-\[ \psi^{(0)} = \psi^{n}, \]
-\[ k^{i} = f(t^{n} + \Delta t\,\alpha_{i},\,\psi^{i-1}), \]
-\[ \psi^{i} = \psi^{n} + \Delta t\,\sum^{i}_{j=1}\,\beta_{i+1,j}\,k^{j}, \quad \textnormal{mit} \quad i \in [1,2,...,N] \]
-\[ \psi^{n+1} = \psi^{N}. \]
+\begin{align*}
+ \psi^{(0)} &= \psi^{n}, \\
+ k^{i}      &= f(t^{n} + \Delta t\,\alpha_{i},\,\psi^{i-1}), \\
+ \psi^{i}   &= \psi^{n} + \Delta t\,\sum^{i}_{j=1}\,\beta_{i+1,j}\,k^{j}, \quad \textnormal{mit} \quad i \in [1,2,...,N] \\
+ \psi^{n+1} &= \psi^{N}. 
+\end{align*}
 }}}
 The coefficients can be written in a so-called Butcher-Tableau:
@@ -31 +33 @@
 {{{
 #!Latex
-\[ \hat\psi_{1} & = & \psi_{n} + \frac{1}{3} \Delta t f\left(\psi_{n}\right) \]
-\[ \hat\psi_{2} & = & \hat\psi_{1} + \frac{1}{48} \Delta t \left( 45 f\left(\hat\psi_1\right) - 25 f\left(\psi_{n}\right) \right) \]
-\[ f\left(\hat\psi_{1}\right) & = &-153 f\left(\hat\psi_{1}\right) + 85 f\left(\psi_{n}\right) \]
-\[ \hat\psi_{3} & = & \left( \psi_{n+1} \right) = \hat\psi_{2} + \frac{1}{240} \Delta t \left( 128 f\left(\hat\psi_2\right) + 15 f \left(\hat\psi_{1}\right) \right) \]
+\begin{align*}
+ \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) \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) + 15 f \left(\hat\psi_{1}\right) \right) 
+\end{align*}
 }}}
 
 For reasons of clarity the [../../app/inipar#timestep_scheme time integration] for several schemes (further schemes are: Leapfrog, Euler and 2^nd^ order Runge-Kutta scheme) is implemented as follows (here e.g. the u-component of velocity):
 
+
 {{{
-#!Latex
-\begin{split}
- \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) * \textnormal{ddx} \ \ ) \\
-                             - \textnormal{tsc}\left(5\right) * \textnormal{rdf}\left(k\right) * \left(\textnormal{u}\left(k,j,i) - \textnormal{ug} \right) 
-\end{split}
+u_p(k,j,i) = ( 1.0 - tsc(1) ) * u_m(k,j,i) + tsc(1) * u(k,j,i) + dt_3d * ( 
+               tsc(2) * tend(k,j,i) + tsc(3) * tu_m(k,j,i)
+             + tsc(4) * ( p(k,j,i) - p(k,j,i-1)) * ddx )
+             - tsc(5) * rdf(k) * ( u(k,j,i) -ug )
 }}}
+
 
 and steered by the array {{{tsc(1:5)}}}