STELLA and iThink provide capabilities to model spatial problems. Version 9.1.2 expands these capabilities to allow more intuitive equation formulations.
Spatial modeling is concerned with modeling behavior over space as well as time. Common applications include modeling flows in lakes and streams or modeling development in an urban landscape. To demonstrate the new capabilities in version 9.1.2, the one-dimensional diffusion problem will be modeled in STELLA.
Consider the diffusion of heat through a one meter metal bar when a constant temperature is applied to both ends. This problem is typically introduced as the following partial differential equation for temperature function u(x, t) with the given boundary conditions:
∂u/∂t – k ∂2u/∂x2 = 0
u(x, 0) = 0, 0 < x < 1
u(0, t) = u0, t ≥ 0
u(1, t) = u0, t ≥ 0
Here, k is the diffusion coefficient (based on thermal conductivity, density, and heat capacity) and u0 is the constant temperature applied to both ends of the bar. The finite difference solution to this problem (with the above initial conditions) is:
u(x, t + dt) = u(x, t) + ∂u/∂t
∂u/∂t = k (u(x + dx, t) – 2u(x, t) + u(x – dx, t))/dx2
It is also possible to derive a closed-form solution. Unless you are an astrophysicist, this solution probably does little to help you understand what is happening. It is far more intuitive to develop a physical model of the underlying mechanics than to wrestle with the mathematics.