Sample Problems > Usage > 1D > 1d

{ 1D_CYLINDER.PDE

This problem tests the implementation of 1D cylindrical coordinates in FlexPDE.

A distributed source is applied to a heatflow equation. The source is chosen as

the analytic derivative of an assumed Gaussian solution. The numerical solution

is then compared to the analytical solution.

}

title '1D Cylinder Test -- Gaussian'

coordinates

cylinder1 { default coordinate name is 'R' }

variables

definitions

k = 1

w=0.1

{ assume a gaussian solution }

u0 = exp(-r^2/w^2)

{ apply the correct analytic source for cylindrical geometry (we could use

div(k*grad(u0)) here, but that would not test the 1D Cylinder expansions) }

s = -(4/w^2)*(r^2/w^2-1)*u0

left=point(0)

right=point(1/10)

equations

U: div(K*grad(u)) +s = 0

boundaries

region 1

start left point value(u)=u0

line to right point load(u)=(-2*k*r*u0/w^2)

monitors

elevation(u) from left to right

plots

elevation(u,u0) from left to right

elevation(u-u0) from left to right as "Error"

elevation(-div(grad(u)),s) from (0.01) to right

elevation(-grad(u),-grad(u0)) from (0.01) to right

end

1d_cylinder