ORIGINAL: berthier
"If you want to simulate a turn, you can approximate f(t+dt) by f(t)+f'(t)*dt. dt is one turn. f'(t) is the slope of f measured at time t. This is called explicit integration. Or you can approximate it by f(t)+f'(t+dt)*dt. f'(t+dt) is the slope of f measured at time t+dt. That's called implicit integration. Or you can approximate it by f(t)+0.5*(f'(t)+f'(t+dt))*dt. That's Crank-Nicholson. Explicit integration is unstable--it tends to diverge from reality. Implicit integration is always stable, even when it shouldn't be. Crank-Nicholson is a compromise. Every game I know of uses explicit integration. Unless you keep them on the tracks somehow, they're likely to go off in weird and wonderful directions. "
Sorry, and I really dont mean to offend, but this is an entirely meaningless explanation to me and I assume to many others who haven't done higher maths...and I do have two degrees in science/health science and still have no idea what point you are trying to make. Perhaps an example of each case rather than a formula might help?
My guess...
Game designer's need game data to check things, but they can't spend all the time playing the game out to get varied data to check various situations. So they 'simulate' game results at various stages using formulae to get approximate results quickly that they can use to check how they think the game will turn out.
"If you want to simulate a turn, you can approximate f(t+dt) by f(t)+f'(t)*dt. dt is one turn. f'(t) is the slope of f measured at time t. This is called explicit integration.
f(t+dt) is the function (or game state) at the turn desired. t is a known turn result, and dt is the number of additional turns required to get to the unknown turn result.
For example, you can simulate the results of turn 30 using the results of turn 20 + 10 extra turns worth of functioning. t=20, dt =10, the result at turn 30 will be simulated by taking the result of turn 20 and adding 10 estimated (the estimation is what f is all about) turn results to that.
The estimation process here is done by looking at turn results graphically (or mathematically) and finding the slope of the graph (f'(t)). For example, if a german infantry div lost 5% of its strength during turn 20, the estimation is that it will lose 5% of its strength in each of the following turns (vastly oversimplifying here). So the division's state at turn 30 will be estimated as its state at turn 20 (
f(t)) plus the slope it was on at turn 20 (
f'(t)=-5%) multiplied by the number of turns (
dt=10). The division will have lost 50% strength in 10 turns (or be at .95^10 strength = approximately 60% strength, depending on your model).
This nearly always produces 'wrong' results (diverges from reality, or is unstable) because every existing result includes some random factor, or noise. So that measured 5% losses should maybe have been 4.2% on average, but the Russians got a good roll or something. This minor fault in the estimation process multiplies itself out each additional turn, making a bigger and bigger fault, not to mention introducing additional errors (after losing only 42% strength instead of 50% strength the division is fighting over 20% stronger, so it's current calculations are now 20% off, and they've been off all the way through).
That's explicit integration whatsit.
Implicit integration is exactly the same except you use the 'end' slope (at t+dt, or turn 30) instead of the 'start' slope (at turn 20). I don't remember why it's stable, but it doesn't give any more accurate results because the end slope is just as affected by a single random noise variable. Maybe the end slope is 3.8% loss (true average value 4.2 remember), so instead of over calculating losses all the way we will under calculate the losses all the way.
The Crank-Nicholson method uses an estimated average slope by adding together the starting slope and the finishing slope and dividing by 2 (multiplying by 0.5). Hence it is more likely to give better results because 2 random noise variables have a higher chance of canceling each other out than multiplying each other. 0.5(5+3.8)= 4.4% losses used in the calculation - closer than either other value.
It won't always be closer, but it usually will, and done many times in multiple calculations (ie that 10 turn break will usually be done 1 turn at a time) it will be closer very much more often than not - the distribution of an average of two values is much tighter around the mean than the distribution of individual values.
I've done my best to forget this stuff over the last 20 years, so sorry if there re some inaccuracies there but that's what it looks like to me in as layman's terms as I can think of..
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