Changes between Version 11 and Version 12 of doc/tec/rk3
- Timestamp:
- Jan 11, 2011 1:23:03 PM (14 years ago)
Legend:
- Unmodified
- Added
- Removed
- Modified
-
doc/tec/rk3
v11 v12 1 1 = 3^rd^ order Runge-Kutta scheme = 2 2 3 For the di cretization in time a 3^rd^ order low-storage Runge-Kutta scheme with 3 stages recommended by Williamson (1979) is used.3 For the discretization in time a 3^rd^ order low-storage Runge-Kutta scheme with 3 stages is used recommended by Williamson (1979). 4 4 Generally an N-stage Runge-Kutta scheme discretizes an ordinary differential equation of the form 5 5 {{{ … … 15 15 \[ \psi^{n+1} = \psi^{N}. \] 16 16 }}} 17 The coefficients can written in a Butcher-Tableau in the following way:17 The coefficients can be written in a so-called Butcher-Tableau: 18 18 || α,,1,, || β,,1,1,, || {{{0}}} || ... || || || 19 19 || α,,2,, || β,,2,1,, || β,,2,2,, || {{{0}}} || ... || || … … 22 22 || || β,,N+1,1,, || β,,N+1,2,, || ... || β,,N+1,N-1,, || β,,N+1,N,, || 23 23 24 The appendant coefficients for the used Runge-Kutta scheme reads:24 The appendant coefficients for the applied Runge-Kutta scheme reads: 25 25 || {{{0}}} || {{{0}}} || {{{0}}} || {{{0}}} || 26 26 || {{{1/3}}} || {{{1/3}}} || {{{0}}} || {{{0}}} || … … 28 28 || || {{{1/6}}} || {{{3/10}}} || {{{8/15}}} || 29 29 30 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 (thereforlow-storage scheme) as follows:30 To save storage 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 (therefore low-storage scheme) as follows: 31 31 {{{ 32 32 #!Latex … … 37 37 }}} 38 38 39 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 follow ing (here for examplethe u-component of velocity):39 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): 40 40 41 41 {{{ … … 56 56 || {{{1}}} || {{{8/15}}} || {{{1/15}}} || {{{0}}} || {{{1}}} || 3^rd^ substep 57 57 58 {{{u_p}}} is the prognosticated and {{{u}}} the current velocity at each substep. {{{u_m}}} denotes the velocity of the lastsubstep (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].58 {{{u_p}}} is the prognosticated and {{{u}}} the current velocity at each substep. {{{u_m}}} denotes the velocity of the previous 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]. 59 59 60 60 … … 64 64 * '''Baldauf, M., 2008:''' Stability analysis for linear discretisations of the advection equation with Runge-Kutta time integration. ''J. Comput. Phys.'', '''227''', 6638-6659. 65 65 66 * '''Durran, D. R., 1999:''' ''Numerical methods for wave equations in geophysical fluid dynamics.'' Springer Verlag, New York, 1. Auf \/l., 465 S.66 * '''Durran, D. R., 1999:''' ''Numerical methods for wave equations in geophysical fluid dynamics.'' Springer Verlag, New York, 1. Aufl., 465 S. 67 67 68 68 * '''Williamson, J. H., 1980:''' Low-storage Runge-Kutta schemes. ''J. Comput. Phys.'', '''35''', 48-56.
