To stay or leave PhD after having completed 4 years I apologize if this is too long a post, but I could really do with a few
pointers about my current situation.

I am 25 years old and I will complete 4 years of my PhD in a computational applied mathematics  program in the US in August 2013.

My bachelors was in pure mathematics.  I had gotten interested in numerical analysis in that time and so I had applied to my current PhD program. I have been under my adviser for 3 years now (the first year at my university is spent in coursework).  A PhD at my university is usually 5 years long. 

Right now, I have almost nothing to report in the way of research, and consequently no publications , no conference submissions. I am getting increasingly nervous and frustrated about whether or not I will make it, even if I give myself an extra year by funding myself. 

My adviser has consistently been making me work on uninteresting stuff, where most of the work involved is purely technical like writing brain-dead code, with almost zero chance for innovation. 

BREAKDOWN OF MY PHD 

After monkeying around reading research papers, in the first year under my adviser,  he got very confident about getting an industrial project and got me working on that, in anticipation that the contract would go through. At the end of the year we found out that we did not get the project.  

In the second year, he said he wanted to get into GPU parallel computing 
and to implement a few fluid dynamics algorithms. I slogged over many manuals, spent months and months writing and debugging code, all the time thinking that 
this would be used to do some simulations he was interested in and get them published. But at the end of the second year my professor completely lost interest in these numerical techniques he was making my implement. 

Seeing his capricious attitude, I almost wanted to quit then and there itself. But I decided to just stick it out, thinking it might be 'just a phase'.  Due to funding issues, he once more got me working in the third year on another project which essentially involved writing a lot of stupid code, and running endless benchmark tests.  

I have basically ended up trying to do a PhD in mathematics without any mathematics in this PhD. 

Finally, a couple of weeks back, I told him that I had had enough, and to give me some actual 
problems/material to work with. After about an hour of discussion, and informing him that I was ready to fund my self if required, he finally gave me a couple of possible starting points for what I hope would actually turn out to be worthwhile research.

MY QUESTIONS


I do realize it was extremely foolish waiting for so long before putting my foot down, and not having the courage to speak up before. My adviser is well-regarded by colleagues in his field, and maybe I was subconsciously scared of 
contradicting his handling of my PhD for pissing him off. 

But even though he has now suggested problems which do seem interesting, after having had so many negative experiences  I am very skeptical about the future. How should I proceed, and what are the factors I should consider ? 

Frankly, I am feeling very burned out.  In the way of future plans, I have been toying with the idea of dropping out,  getting a break for a few months and then sitting for some entrance exams for a Masters in Economics in some good universities back in my home country. I always found economics very interesting through my undergrad and more so these past few-months while studying it has a hobby.
Continuing would require me to stay on for an extra year till August 2015, 
which leaves me about 17 months tops from now, before I start hunting in academic job market. This includes about 2-3 months I will have to spend doing literature review on the proposed topics and learning the requisite mathematical tools. 

So if I decide to stay on, how should I re-structure my study/research time and the relationship with my adviser in these 17 months so that I can make some head-way.


Maybe 17 months is too short a time? Any suggestions would be really helpful !!
Firstly, no apology needed, your question is thorough and easy to read and understand.  It sounds like you are in quite an unpleasant situation.

Don't take any advice I may give as gospel, but in answer to your questions:

You are most certainly not foolish to wait until now to stand up to your advisor, you have had several leads and have given many chances for the project to kick into gear. You have every right and reason to feel skeptical about the current promises and project direction.

Ultimately, how you proceed is up to you (you're probably understandably sick of hearing that), but look at the following considerations (no doubt many other members will add to this):


The new direction could well be a winner, leading to papers, conferences and most of all, fulfillment.  It could also be a good one as now, you have made your feelings clear to your supervisor.
Could this be just another academic 'false positive'?


Perhaps outline a couple of potential papers and present them to your advisor (this is something I do).  This could be an ongoing thing, alongside your research - outline potential papers.

As for the timeframe, 17 months - I would not be too worried about that - I have been able to get three papers published in less than ten months, with a 4th on the way and the 5th planned (I finish my Ph.D. at the end of the year).

I hope this helps, and I hope it all gets sorted out for you.