Re: Numerical solution of the heat equation on a disk

*To*: mathgroup at smc.vnet.net*Subject*: [mg113619] Re: Numerical solution of the heat equation on a disk*From*: Francois Fayard <FFayard at slb.com>*Date*: Fri, 5 Nov 2010 05:10:27 -0500 (EST)

Thank you for that explanation. I was missing a few points here. Regards, Francois On Nov 4, 2010, at 7:43 AM, Mark McClure wrote: > On Thu, Nov 4, 2010 at 4:58 AM, Francois Fayard <FFayard at slb.com> wrote: > >> I would like to get a numerical simulation of the heat equation with >> Dirichlet boundary conditions on a disk. With the problem I have, >> the function does not depend on theta, so we get : >> u_t == u_rr + (1/r) u_r >> >> It introduces a singularity as goes to 0 and Mathematica can not >> solve the problem with NDSolve. Is there a way to go around this? > > Of course, there is no singularity, rather a removeable discontinuity. > D[u,r]/r, for example, can be replaced with D[u,r,r] at r==0. We can > define some auxilliarly functions so that the equation takes the > proper form at zero. We also use the rotational symmetry of the > solution to obtain the boundary condition at zero, nameley D[u,r]==0. > Putting this all together, we get something like so: > > f1[r_ /; r > 0] :== 1; > f1[r_ /; r ==== 0] :== 2; > f2[r_ /; r > 0] :== 1/r; > f2[r_ /; r ==== 0] :== 0; > Clear[u]; > {sol} == NDSolve[{Derivative[0, 1][u][r, t] ==== > f2[r]*Derivative[1, 0][u][r, t] + > f1[r]*Derivative[2, 0][u][r, t], u[r, 0] ==== 1 - r^2, > Derivative[1, 0][u][0, t] ==== 0, u[1, t] ==== 0}, > u[r, t], {t, 0, 2}, {r, 0, 1}]; > u[r_, t_] == u[r, t] /. sol; > pics == Table[ParametricPlot3D[{r*Cos[th], r*Sin[th], u[r, t]}, > {r, 0, 1}, {th, 0, 2 Pi}, PlotRange -> {-1, 1}], > {t, 0, 0.3, 0.01}]; > ListAnimate[pics]