Intro to Ascending
This isn't going to provide a step-by-step of how to ascend, or jumar, or "jug" up a rope; but rather focus on what's involved with the whole process. It is best practice for any canyoneering adventure, that at least one member in your group has the confidence and ability to ascend a rope, in case that rope gets stuck.
There are numerous ways to ascend a rope, but the two primary ways that canyoneers do it is done by tiblocs and handled ascenders
Tiblocs are incredibly light and are made by a renowned climbing company, Petzl. They work on rope sizes from 8 to 11 mm, and work even in muddy or icy situations.
They use a carabiner which cinches down the device to the rope which ensures that it will grab the rope. It only works in one direction. Once you move it up, you can't move it back down, unless you take off the carabiner and reposition it lower on the rope.
At heavy loads while using Tiblocs, it WILL damage the rope. At those times, these are considered to be for "emergency use"
Here's a video of a simple demonstration of the device in action.
The "best" device in my opinion for ascending, is one that also does NOT damage the rope.
Image Credit: blackdiamondequipment.com
Petzl makes one called the "Ascension".
Black Diamond also makes a very similar version called the "Index Ascender". In fact, I prefer this one for myself because it is a little bigger and easier to hold on to. Your mileage may vary!
These do NOT damage the rope.
To quote from their website:
"In these situations with fall factor less than one, the ascender's teeth do not damage the rope.
Despite their aggressive appearance, the teeth on the BASIC or ASCENSION are not harmful to the rope in normal use.
The spring pushes the cam against the rope and causes it to engage the rope without sliding (and therefore without tearing), especially if the teeth are new and pointy."