Re: Checking the Results of NDSolve

*To*: mathgroup at smc.vnet.net*Subject*: [mg37313] Re: Checking the Results of NDSolve*From*: atelesforos at hotmail.com (Orestis Vantzos)*Date*: Wed, 23 Oct 2002 02:56:44 -0400 (EDT)*References*: <ap07cn$c4f$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

If there is a preserved quantity in the ODE system (and the residuals referred to a previous post on this thread certainly are) you can use it to perform an independent check. It must be noted though that the residuals are not appropriate for an entirely independent check in most cases, since most numerical solvers use them for exactly this kind of check. ODEs from physics usualy have one or more 'integrals'(preserved quantities); typicaly the energy of the system. Example: the ODEs {x'[t]==-y[t],y'[t]==x[t]} describe a rotation around the origin with fixed angular velocity. The preserved quantity is ofcourse the distance from the origin d[t]=Sqrt[x[t]^2+y[t]^2]. One could use Abs[d[0]-d[t]] as an independent check for the validity of the approximate NDSolve solution. My advice is study carefuly the ODEs before feeding them to NDSolve; analytical results (such as the existence of preserved quantities) can radicaly improve the usefulness of numerical results. Orestis