This method is derived from Per's 'cage' method. I like this method a lot but my biggest problem with that was the centres at the end, so I stuck this part nearer the start when you have more freedom. It relies on commutators quite heavly so I will try and show you some examples with each picture when needed. A nice advantage is that you only have one parity (you can even use a parity alg from the centres-edges-3x3x3 method, the OLL one) and that you don't have a 3x3x3 stage at the end, which is a plus for all the corners first people :)
I invented this while coming back from a cube meetup. My times were around 3:00 (using the cage method) and using this they dropped to around 2:00 with me breaking my PB (set with a centres-edges-3x3x3 method) of 1:43 with about an hour after it was conceived.
Lower case letters denote single layer turns.
Here we have a scrambled cube. The first thing we are going to do is make a 'Roux Block'. By this, I mean create a 1x3x4 block like in the picture below. I usually do this by getting a centre (white) and just adding the edges like you're making one layer. This is quite trivial to do and requires little thinking. I build the same block for better colour recognition later on, but neutrality may help in some lucky cases and opposite colour solving. I usually build the block on L.
Once we have the 1x3x4 block done, we move on and solve the centres. I use Stefan Pochmann's method for this part, but execute it differently generally. Firstly I keep the 1x3x4 block on L and solve the centres in the l layer/ring. This is practically the same as stefan's method, but make sure you only use <l,r,R,U> to solve them as to preserve the 1x3x4 block. Then when doing the rest of the centres, I place the 1x3x4 block on D with the free part in the B layer. I shoot the centres from U onto the B face with LlU'Ll' and Rr'URr. An advantage for this is that you can see pretty much every edge apart from the one you are solving, so it is good for looking ahead. When you're going to solve a face, always move it to B then shoot the centres in U to B while looking at the next centre you're going to put in. If the U centres are solved and the rest of the cube isn't, simply get a dodgy centres from the E slice into U with one of the moves or directly solve it.
Before you start this step, replace the 1x3x4 block back to L by rotating the cube if it isn't already there.
The 'Waterman' step.
In this stage we will solve all the edges pieces in the M layer leaving only the final layer on R left to complete. I do this with a combination of 3x3x3 algorithms and commutators. There will be lots of little things that you can discover yourself to make yourself go faster. I will present some examples and leave you to do the rest. I advise against learning these. Instead, learn how each one works and what it does, so you can create your own to do different things.
In this step we are going to solve the edges of the last layer. This is the area of the I think I can improve on the most and optimise, personally. I suggest you learn these two following algorithms, as they are very useful; RUR'U'rR'URU'r' and rUR'U'r'URU'R'. Again, we solve this step with commutators. Because I'm not a commutator god, I try to use the two stated algorithms and some 3x3x3 algorithms to limit the number of commutator 3-cycles of edges you have to do to a minimum. I do one or two a solve this way. Gradually though, I am finding that recognition is speeding up and the number of three cycle cases I have to 'know' are relatively small. I will try and explain the concepts behind creating them as best I can, but I find it difficult. I learnt how to do them from Per, and I think he would be more than happy to talk about them.
There is only one parity in this method (because we are directly solving, the permutation parity doesn't really exist), but it's quite difficult. It comes up 50% of the time. You may end up with just two edges swapped. if this happens, you can use an OLL parity that just flips a single edge to solve adjacent edges, and l'U2 l'U2 x U2l' U2r x' U2r' U2l2 to solve the case presented in the image. when a case comes up not like the one shown, you can use setup moves combined with one of the parities to solve it, or an alternative would be to use the quick 3-cycle algorithms from the last step to setup and do the inverse to finally fix the puzzle.
An alternative approach proposed is to solve the yellow centre after the white, then you complete the 1x3x4. This doesn't hamper your ability to build the 1x3x4 at all, and makes the centres extremely easy to solve (I used to hate the centres part, now it's much faster) as you can keep the 1x3x4 on L (less cube rotations), solve them like in the centres-edges-3x3x3 and not have to worry about breaking the 1x3x4 as much.