Changes between Version 8 and Version 9 of doc/tec/rk3

Timestamp:

Jan 10, 2011 3:01:06 PM (14 years ago)

Author:

suehring

Comment:

--

doc/tec/rk3

@@ -16 +16 @@
 }}}
 The coefficients can written in a Butcher-Tableau in the following way:
-|| α,,1,, || β,,1,1,, || 0 || ... || || ||
-|| α,,2,, || β,,2,1,, || β,,2,2,, || 0 || ... || ||
+|| α,,1,, || β,,1,1,, || {{{0}}} || ... || || ||
+|| α,,2,, || β,,2,1,, || β,,2,2,, || {{{0}}} || ... || ||
 || ...    ||  ... || ||  ||  || ||
-|| α,,N,, || β,,N,1,, || β,,N,2,, || ... || β,,N,N-1,, || 0 || 
+|| α,,N,, || β,,N,1,, || β,,N,2,, || ... || β,,N,N-1,, || {{{0}}} || 
 ||        || β,,N+1,1,, || β,,N+1,2,, || ... || β,,N+1,N-1,, ||  β,,N+1,N,, || 
 
 The appendant coefficients for the used Runge-Kutta scheme reads:
-||   0  ||   0   ||   0   ||   0  ||
-||  1/3 ||    1/3 ||    0  ||  0   ||
-|| 1/2  ||  -3/16 || 15/16 ||  0  ||
-||      ||  1/6  ||  3/10 || 8/15 ||
+||   {{{0}}}  ||   {{{0}}}   ||   {{{0}}}  ||   {{{0}}}  ||
+||  {{{1/3}}} ||    {{{1/3}}} ||    {{{0}}}  ||  {{{0}}}   ||
+|| {{{1/2}}}  ||  {{{-3/16}}} || {{{15/16}}} ||  {{{0}}}  ||
+||      ||  {{{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 after the second substep to save storage (therefor low-storage scheme) as follows: 
@@ -37 +37 @@
 }}}
 
-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):
+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 following (here for example the u-component of velocity):
 
 {{{
@@ -44 +44 @@
  \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(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}
 }}}
 
-and steered by the array tsc(1:5)
+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
+|| {{{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
 
-
-
-Sorry for incompleteness. A further description will follow in the next days.
+{{{u_p}}} is the prognosticated and {{{u}}} the current velocity at each substep. {{{u_m}}} denotes the velocity of the last substep (needed for Leapfrog). {{{tend}}} is the current tendency and {{{tu_m}}} the combined tendencies of the prior substeps. {{{tsc(4)}}} steers the preconditioning of the [../../app/inipar#psolver pressure solver] and {{{tsc(5)}}} the [../../app/inipar#rayleigh_damping_factor rayleigh damping].