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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