NonLinODE

{ NONLINODE.PDE }

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

This example shows the application of FlexPDE to the solution of a

non-linear first-order differential equation.

A liquid flows into the top of a reactor vessel through an unrestricted

pipe and exits from the bottom through a choke value.  This problem is

discussed in detail in Silebi and Schiesser.

This is a problem in viscous flow:

    dH/dt = a - b*sqrt(H)

The analytic solution satisfies the relation

    sqrt(H0) + (a/b)ln[a-b*sqrt(H0)]

     - sqrt(H) - (a/b)ln[a-b*sqrt(H)] = (b/2)*t

which can be used as an accuracy check.

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

title

"NONLINEAR FIRST ORDER ORDINARY DIFFERENTIAL EQUATION"

select

ngrid=1               { Since there is no spatial information required,

                         use the minimum grid size }

variables

Height(threshold=1)   { declare Height to be the system variable }

definitions

a = 2                 { define the equation parameters }

b = 0.1

H0 = 100

                       { define the accuracy check }

T0 = sqrt(H0) + (a/b)*ln(a-b*sqrt(H0))

Tcheck = sqrt(Height) + (a/b)*ln(a-b*sqrt(Height))

initial values

Height = H0

equations

dt(Height)  = a - b*sqrt(Height)    { The ODE }

boundaries

region 1              { define a fictitious spatial domain }

   start (0,0)

   line to (1,0) to (1,1) to (0,1) to close

time 0 to 1000          { define the time range }

histories

                       { Plot the solution: }

history(Height) at (0.5,0.5)

                       { Plot the accuracy check: }

history([T0 - Tcheck - (b/2)*t]/[(b/2)*t]) at (0.5,0.5)

               as "Relative Error"

end