Normally, cooking a stew (not counting prep) in less than a three hours seems like rushing it to me. I can't imagine that you'd ever get the fall-apart tender meat that people expect from a stew, but...
There are a number of techniques that I've seen professional cooks use when they're in a time crunch that may help you overcome time constraint problems such as this one:
- Cut the ingredients into smaller pieces
- Use larger bottomed or multiple pans (even electric ones as mentioned in the comments)
- Use a deep fryer (not recommended for a stew ;-), though it would certainly help get the meat up to temperature)
- Use a microwave (works well for some ingredients, not so much for others)
- Use a pressure cooker (if available, this is what I would recommend for a stew)
From a scientific perspective, it's going to take 5 times as much energy to bring 20 portions worth of ingredients to temperature as it would 4 portions. Coming up with an exact formula would require some detailed information such as the BTUs put out by the stove, the spread of the burner, and surface area, shape, and material of the pan being used, etc. However, we do know that a lower portion of the energy will be lost when cooking 20 portions due to more surface area of the pan being in contact with the ingredients (even if only on the sides). I would guess that the wasted energy probably ranges from 50% if you're cooking small portions to 20% if you're cooking large portions (again, depending on the efficiency of your cooking arrangement, which will vary widely).
F is the energy required to bring 4 portions of the food to temperature, and
S is the energy per time unit put out by the stove, and
t1 is the time required to bring 4 portions up to cooking temperature, and we guess that we have 50% efficiency when cooking 4 portions, we have
F=.5*S*t1. When we increase to 20 portions, assuming the efficiency increases to 80%, we'd have
5F=.8*S*t2. Solving for
t2 relative to
t1, we get
t2=3.125*t1, so (given the assumptions of course), bringing 20 portions up to cooking temperature would take 3.125 times as long as bringing 4 portions up to temperature. To speed things up, you would have to alter the input energy (more burners, electric assistance, etc.), or increase the efficiency (more pan surface area, smaller cut ingredients). A more general equation would be
Eb being the time to cook a baseline amount, its mass, and the efficiency for that volume, and
E being the time for the new amount, its mass, and its efficiency.
Of course this is just the time to bring the ingredients up to cooking temperature. Once there, the volume of ingredients doesn't matter much unless there is a lack of convection due to the thickness during the cooking phase, so with more portions you may need to stir more (this would be the solution to the gradient problem). After coming to cooking temperature, adding energy faster or more efficiently won't help much (if at all), but changing the pressure will change how quickly the food cooks, which is why this is what I would recommend. The cooking phase is the majority of the time, so a 10% savings there will save you more than a 10% savings during the 'coming up to temperature' phase. Depending on the stew, you may or may not need some time at the end for reduction and thickening as well.
In addition to all this, if you're doing prep work during the time you've listed, I'd skip the mise en place and get whatever you can heating up immediately so you can get as much heat as possible into as much of the stew as possible as quickly as possible.