<p>Yeah, I know the title is a mouthful...</p>

<p>What I mean is to say is how do you communicate with a subject matter expert who needs a theory coded and tested?</p>

<p>For example, weather simulation is a collaboration between meteorologists, computer scientists, and software engineers.  The computer scientists and software engineers generally speak the same language, but he meteorologist is in a completely different world.</p>

<p>How do you increase the level of communication and understanding between disciplines?  And not necessarily just for weather, other sciences too.</p>

The shortest possible answer is Continuous customer involvement.
All the pretty UML diagrams, crayola UI mockups, explanations-to-four-year-olds and other techniques will never give the full experience of using a working application. Keeping the consumer in the loop allows for a feedback cycle both to the client, and from the client to you.  This symbiotic relationship has the greatest likelyhood of producing a product that will be useful to them.
If you go into a box and come out with a product that you think they need it will likely be a whole lot of what they don't want.  By regularly demoing your product you limit the impact of any misunderstanding, so that you don't spend too much time going down the wrong path.
It can be compared to dead reckoning. If you blindfold yourself and try to navigate through an area you know, the error between where you are and where you think you are will accumulate with time.  If, however, you take off the blindfold periodically you can update your mental location.  There will still be an error factor, but you are eliminating the accumulating error factor.
Even if you think your communication/explanatory skills are top notch, you still have to account for error in the way they communicate.