Sample Problems > Usage > ODE

{ 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.

Since FlexPDE requires a spatial domain, we solve the equation on

a simple box with minimum mesh size.

}

title

"NONLINEAR FIRST ORDER ORDINARY DIFFERENTIAL EQUATION"

select

{ Since there is no spatial information required, use the minimum grid size }

ngrid=1

variables

{ declare Height to be the system variable }

Height(threshold=1)

definitions

{ define the equation parameters }

a = 2

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 { The ODE }

Height : dt(Height) = a - b*sqrt(Height)

boundaries

{ define a fictitious spatial domain }

region 1

start (0,0)

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

{ define the time range }

time 0 to 1000

plots

for t=0, 1, 10 by 10 to endtime

{ 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

nonlinode

nonlinode