current in wire with FDTD

Hi,

I have a relative simple question. How do I simulate/show a DC current in a wire/resistor with FDTD?

My approach was to create a 3D free space volume and adapt a part to create a wire (with some (positive) conductivity). Then I assigned initial E-fields to both ends (e.g. 1 and 0 V). What happens now is that the E-field decreases with 1/e in the wire, resulting in an equivalent decreasing current (density, J). But this is not what I want, since in practice a constant current will flow through the wire when a voltage source is connected.

Can someone explain how I should adapt my model? I guess I have to implement a static term or so.

I would use a constant hard source. After each iteration reset the E-field to a fixed term.
That way the path integral of the electric field along the wire remains constant.

Variations:
a) use only one cell along the wire
b) distribute the volatge evenly over the wire
c) use a fixed current (reset the magnetic fields in a loop around the wire to a constant term)

I don't know if there are any other methods. Please let me know what worked best in the end.