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{  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

   Temp : 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