Version 8 (modified by suehring, 14 years ago) (diff)

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3rd order Runge-Kutta scheme

For the dicretization in time a 3rd order low-storage Runge-Kutta scheme with 3 stages recommended by Williamson (1979) is used. Generally an N-stage Runge-Kutta scheme discretizes an ordinary differential equation of the form

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"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./ab9a045fbec844e76eb011e3d45b3e3a7e5d0f24.tex\nLaTeX2e <2011/06/27>\nBabel <3.9h> and hyphenation patterns for 78 languages loaded.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/size10.clo))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/inputenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/utf8.def\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cmap/cmap.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/cm-super/type1ec.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1cmr.fd))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/fontenc.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap\n>>) (/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amstext.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/tools/bm.sty)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/preview.sty\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/preview/prtightpage.def)\n)\nNo file ab9a045fbec844e76eb011e3d45b3e3a7e5d0f24.aux.\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n<<ot1.cmap>><<oml.cmap>><<oms.cmap>><<omx.cmap>>\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/localdata/software/texlive/2013/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Misplaced alignment tab character &.\nl.14 \\[  \\frac{d \\psi}{d t} &\n                              = & f(t,\\psi) \\]\n! Misplaced alignment tab character &.\nl.14 \\[  \\frac{d \\psi}{d t} & = &\n                                  f(t,\\psi) \\]\nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}] (./ab9a045fbec844e76eb011e3d45b3e3a7e5d0f24.aux) )\n(see the transcript file for additional information)</localdata/software/texliv\ne/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></localdata/softwar\ne/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb>\nOutput written on ab9a045fbec844e76eb011e3d45b3e3a7e5d0f24.pdf (1 page, 21457 b\nytes).\nTranscript written on ab9a045fbec844e76eb011e3d45b3e3a7e5d0f24.log.\n"

as follows ( Baldauf, 2008 ):

\[ \psi^{(0)} = \psi^{n}, \]
\[ k^{i} = f(t^{n} + \Delta t\,\alpha_{i},\,\psi^{i-1}), \]
\[ \psi^{i} = \psi^{n} + \Delta t\,\sum^{i}_{j=1}\,\beta_{i+1,j}\,k^{j}, \quad \textnormal{mit} \quad i \in [1,2,...,N] \]
\[ \psi^{n+1} = \psi^{N}. \]

The coefficients can written in a Butcher-Tableau in the following way:

α1 β1,1 0 ...
α2 β2,1 β2,2 0 ...
... ...
αN βN,1 βN,2 ... βN,N-1 0
βN+1,1 βN+1,2 ... βN+1,N-1 βN+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

For the implementation it is advantageous to compute ψN from the intermediate solutions ψ1 and ψ2 and combine the local tendencies in one array after the second substep to save storage (therefor low-storage scheme) as follows:

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For reasons of clarity the time integration for several schemes (further schemes are: Leap-frog, Euler and 2nd order Runge-Kutta scheme) is implemented as following (here for example the u-component of velocity):

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and steered by the array tsc(1:5)

tsc(1) tsc(2) tsc(3) tsc(4) tsc(5)
1 1/3 0 0 0 1st substep
1 15/16 -25/48 0 0 2nd substep
1 8/15 1/15 0 1 3rd substep

Sorry for incompleteness. A further description will follow in the next days.