Re: Odd behaviour of solution of PDE

*To*: mathgroup at smc.vnet.net*Subject*: [mg116646] Re: Odd behaviour of solution of PDE*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Tue, 22 Feb 2011 06:24:51 -0500 (EST)*References*: <201102211034.FAA22078@smc.vnet.net> <ijthmr$mn1$1@smc.vnet.net> <ijv26f$73a$1@smc.vnet.net>

Am 22.02.2011 02:06, schrieb Alan Ford: > Hi Oliver, > > I had sent an answer to your question, providing > a simple piece of code, but it seems the post was lost. > > So, rather than submitting the whole matter again, > I would rather formulate my question as a more general issue. > > Suppose you want to integrate a PDE in time and in space, > the range being a<= x<= b. > You have to set boundary conditions. > By mistake, you set one boundary condition not at x = b (the extremum > of integration) > but at x = b + eps (i.e, beyond it). > Mathematica does not complain about it and returns a solution. > My guess was that Mathematica did integrate from x = a to x = b + eps, > in order to match > boundary condition, and then returned solution only in the range > (a,b). > So, no harm, after all. > However, it does not look like the case: in all cases I could > benchmark mathematica > result versus independent solutions, Mathematica simply computes a > wrong solution. > namely, at the first time step it jumps from the initial condition to > a completely wrong value, > and then relaxes with a trend quite similar to the correct one. > > At this time, I'm not asking for hints: it is now quite clear that, > contrary to my belief > in the first post, mathematica is definitely not providing a correct > answer when I supply > the "wrong" boundaries. > But I am wondering why mathematica does not acknowledge that it is > doing an illegal operation The answer is given simply by the fact, that after constructing the matrix of start values at t=0, the step at the boundary is too big to give correct time step estimates there. The difference scheme f_(t+1) (x) = Mean_y h(x, f_t(y)) is used for inner points {x,y}. At the outer boundary f simply stays fixed. The difference quotient for the points next to the boundary are completely wrong estimates then. -- Roland Franzius

**References**:**Odd behaviour of solution of PDE***From:*Alan Ford <fabio.sattin@igi.cnr.it>