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) (./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 ):
![\[ \psi^{(0)} = \psi^{n}, \]
\[ k^{i} = f(t^{n} + \Delta t\,\alpha_{i},\,\psi^{i-1}), \]
\[ \psi^{i} = \psi^{n} + \Delta t\,\sum^{i}_{j=1}\,\beta_{i+1,j}\,k^{j}, \quad \textnormal{mit} \quad i \in [1,2,...,N] \]
\[ \psi^{n+1} = \psi^{N}. \]](/trac/tracmath/22007c852ab1a11e37ad031ac90c189230b8e234.png)
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 to save storage (therefor low-storage scheme) as follows:
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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(./eb8ed16e9ed3c7ba4082ed56cb95aee243f82d33.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 eb8ed16e9ed3c7ba4082ed56cb95aee243f82d33.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 \\[ \\hat\\psi_{1} &\n = & \\psi_{n} + \\frac{1}{3} \\Delta t f\\left(\\psi_{n}\\r...\n\n! Misplaced alignment tab character &.\nl.14 \\[ \\hat\\psi_{1} & = &\n \\psi_{n} + \\frac{1}{3} \\Delta t f\\left(\\psi_{n}\\r...\n\n! Misplaced alignment tab character &.\nl.15 \\[ \\hat\\psi_{2} &\n = & \\hat\\psi_{1} + \\frac{1}{48} \\Delta t \\left( 45 f\\...\n\n! Misplaced alignment tab character &.\nl.15 \\[ \\hat\\psi_{2} & = &\n \\hat\\psi_{1} + \\frac{1}{48} \\Delta t \\left( 45 f\\...\n\n! Missing \\right. inserted.\n<inserted text> \n \\right .\nl.15 ...si_1\\right) - 25 f\\left(\\psi_{n}\\right) \\]\n \n! Missing $ inserted.\n<inserted text> \n $\nl.16 f\\left\n (\\hat\\psi_{1}\\right) & = &-153 f\\left(\\hat\\psi_{1}\\right) + 85 f\\...\n\n! Misplaced alignment tab character &.\nl.16 f\\left(\\hat\\psi_{1}\\right) &\n = &-153 f\\left(\\hat\\psi_{1}\\right) + 85 f\\...\n\n! Misplaced alignment tab character &.\nl.16 f\\left(\\hat\\psi_{1}\\right) & = &\n -153 f\\left(\\hat\\psi_{1}\\right) + 85 f\\...\n\n\n! LaTeX Error: \\begin{preview} on input line 13 ended by \\end{equation*}.\n\nSee the LaTeX manual or LaTeX Companion for explanation.\nType H <return> for immediate help.\n ... \n \nl.16 ..._{1}\\right) + 85 f\\left(\\psi_{n}\\right) \\]\n \n! Missing $ inserted.\n<inserted text> \n $\nl.16 ..._{1}\\right) + 85 f\\left(\\psi_{n}\\right) \\]\n \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/localdata/software/texlive/2013/texmf-var/fonts/map/pdftex/updmap/pdftex.ma\np}]\n! Misplaced alignment tab character &.\nl.17 \\[ \\hat\\psi_{3} &\n = & \\left( \\psi_{n+1} \\right) = \\hat\\psi_{2} + \\frac{...\n\n! Misplaced alignment tab character &.\nl.17 \\[ \\hat\\psi_{3} & = &\n \\left( \\psi_{n+1} \\right) = \\hat\\psi_{2} + \\frac{...\n\n! Missing \\right. inserted.\n<inserted text> \n \\right .\nl.17 ...2\\right) + 16 \\left(\\hat\\psi_{1}\\right) \\]\n \n\n! LaTeX Error: \\begin{document} ended by \\end{preview}.\n\nSee the LaTeX manual or LaTeX Companion for explanation.\nType H <return> for immediate help.\n ... \n \nl.18 \\end{preview}\n \n! Extra \\endgroup.\n<recently read> \\endgroup \n \nl.18 \\end{preview}\n \n(./eb8ed16e9ed3c7ba4082ed56cb95aee243f82d33.aux) )\n(see the transcript file for additional information){/localdata/software/texliv\ne/2013/texmf-dist/fonts/enc/dvips/cm-super/cm-super-t1.enc}</localdata/software\n/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></localdata/\nsoftware/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></lo\ncaldata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.p\nfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfonts/cm/\ncmr10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public/amsfo\nnts/cm/cmr7.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type1/public\n/amsfonts/cm/cmsy10.pfb></localdata/software/texlive/2013/texmf-dist/fonts/type\n1/public/cm-super/sfrm1000.pfb>\nOutput written on eb8ed16e9ed3c7ba4082ed56cb95aee243f82d33.pdf (1 page, 54402 b\nytes).\nTranscript written on eb8ed16e9ed3c7ba4082ed56cb95aee243f82d33.log.\n"
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