Forever

{  FOREVER.PDE

   This problem displays the behaviour of FlexPDE in time dependent problems.

    We posit a field with paraboloidal shape and with amplitude sinusoidal

    in time.  We then derive the source function necessary to achieve this

    solution, and follow the integration for ten cycles, comparing the solution

    to the known analytic solution. }

title 'A forever test'

Variables

    Temp (threshold=0.1)

definitions

    K = 1

    eps = 0

    shape = (1-x^2-y^2)

    Texact = shape*sin(t)

    source = shape*cos(t) - div(K*grad(shape))*sin(t)

Initial values

    Temp = Texact

equations

    div(K*grad(Temp)) + source = dt(Temp)

boundaries

    Region 1

       start(-1,-1)

       value(Temp)=Texact

       line to (1,-1) to (1,1) to (-1,1) to close

time 0 to 20*pi by 0.01

monitors

    for cycle=5

      contour(Temp)              { show the Temperature during solution }

plots                         { write these plots to the .PGX file }

    for t = pi/2 by pi to endtime

      contour(Temp)

      surface(Temp)

      contour(Temp-Texact) as "Error"

      vector(-dx(Temp),-dy(Temp)) as "Heat Flow"

histories

    history(Temp) at (0,0) (0.5,0.5) integrate

    history(Temp-Texact) at (0,0) (0.5,0.5)

end