I'd like to hear from anyone who has built a dialect continuum, in which innovations spreading from different centres affect overlapping subsets of the range (see wave theory). How do you model it? How do you manage the results?

I suppose I'd start with a list of towns and a table determining how likely a town is to adopt an innovation from each of its neighbors. The table can change over time as towns lose or gain influence, or new roads are made.

Then what? Maintain a separate grammar for each town? Impractical.

With a list of sound shifts, you can easily generate a given town's version of a given lexeme as needed; but managing changes in syntax or morphology is, I'd think, not so easy.

I have never tried to do it, but I think the approach you call "impractical" is the way to go. You need to maintain a grammar and a vocabulary for each town and for each period (say, a snapshot every 50 or 100 years) to keep consistency.

The good news that can save a lot of time is, that you probably don't need complete reference grammars for each step in the evolution: Start with a good grammar of the oldest stage of your language continuum and than write the grammars of the evolving dialects/languages in terms of differences to the old grammar.

Only when accumulated differences become too large to be handled well, write a full grammar again.

Yes, having a separate grammar for each town would be the way to go.

When modelling the spread of linguistic difference, do bear in mind geography, rather than just distance. Some regions can be very close in mountainous terrain but still have very different languages/dialects, whereas regions in flat plains (without major rivers) tend to be far more similar.

If you have a map with geographical features, use something like A* to calculate the distance between towns.

As regarding changes in grammar: if your grammar is expressed in the form of rewrite rules, you could allow them to change as well. A fairly radical change would be from

NP -> adj n

to

NP -> n adj

ie the adjective moves behind the noun it modifies. The resulting grammar would of course still be valid. However, you want to make such changes very rare and slow. Alternatively, annotate each rule with a probability, and then have both rules in parallel. Change would then be an increase/decrease in probability.

For language change in general you might be interested in Piotrowski's Law. (Unfortunately the only explanation I have found for it is in German). This models mathematically how change spreads throughout language.

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