Spacetime1

{ SPACETIME1.PDE }

{ **************************************************************

   This example illustrates the use of FlexPDE to solve

   an initial value problem of 1-D transient heatflow as a boundary-value

   problem.  (In most cases, a true initial-value solution is preferable;

   see problem SPACETIME2.PDE).

   Here the spatial coordinate is represented by X, the time coordinate by Y,

   and the temperature by u(x,y).

   With these symbols, the transient heatflow equation is:

       dy(u) = D*dxx(u),

   where D is the diffusivity, given by

       D = K/s*rho,

       K       is the conductivity,

       s       is the specific heat,

   and rho     is the density.

   The problem domain is taken to be the unit square.

   We specify the initial value of u(x,0) along y=0, as well as

   the time history along the sides x=0 and x=1.

   The value of u is thus assigned everywhere on the boundary except

   along the segment y=1, 0<x<1.  Along that boundary, we use the

   natural boundary condition,

              natural(u) = 0,

   since this corresponds to the application of no boundary sources on this

   boundary segment and hence implies a free segment.  This builds in the

   assumption that y = 1 (and hence t = 1) is sufficiently large for steady

   state to have been reached. [Note that since the only y-derivative term is

   first order, the default procedure of FlexPDE does not integrate this term

   by parts, and the Natural(u) BC does not correspond to a surface flux,

   functioning only as a source or sink.]

   This problem can be solved analytically, so we can plot the deviation

   of the FlexPDE solution from the exact answer.

   *************************************************************** }

title "1-D Transient Heatflow as a Boundary-Value problem"

select

    alias(x) "distance"

    alias(y) "time"

variables

    u

definitions

    diffusivity = 0.06         { pick a diffusivity that gives a nice graph }

    frequency = 2              { frequency of initial sinusoid }

    fpi = frequency*pi

    ut0 = sin(fpi*x)           { define initial distribution of temperature }

    u0 = exp(-fpi**2 *diffusivity*y)*ut0       { define exact solution }

Initial values

    u = ut0                    { initialize all time to t=0 value }

equations

    dy(u) = diffusivity*dxx(u) { define the heatflow equation }

boundaries

    Region 1

       start(0,0)

       value(u)=ut0            { set the t=0 temperature }

       line to (1,0)

       value(u) = 0            { always cold at x=1 }

       line to (1,1)

       natural(u) = 0          { no sources at t=1 }

       line to (0,1)

       value(u) = 0            { always cold at x=0 }

       line to close

monitors

    contour(u)

plots

    contour(u)

    surface(u)

    contour(u-u0) as "error"

end