A question about continuing education in an Engineering major or Sciences
It is very possible to go to a PhD program in applied math with a masters or even bachelors degree in engineering. I am in an applied math PhD program and many of my colleagues only took a couple of upper level math classes in college. Applied math is very broad and people with all sorts of backgrounds can get into these sorts of programs. But the question is what can you do now to explore your mathematical interests (an perhaps get into an applied math PhD program too)? The key is to do a few of the following


take a few junior/senior level undergraduate math classes (or if you can handle it, graduate level math courses). You likely have the requisite background to take a class in undergraduate Real Analysis or differential equations. Basically you need to convince an admissions committee that you are capable of writing proofs. You don't need a math degree to do this; you don't even need a math minor to do this. If you do well in a single semester of real analysis some great applied math programs are more than willing to take a gamble on you if you have interesting engineering research and a good personal statement that really explains how you got a passion for math while working in engineering.
incorporate math into your research. You say you do numerics, so prove your method converges, develop analytical error bounds, prove the existence or uniqueness of a solution. If your method is too complex to do the above, or you lack the knowledge to make progress in such an endeavor, create a "toy" problem which is similar but vastly simplified and prove something related to it. Toy problems often lead to stand alone publications, they are interesting to us applied mathematicians in and of themselves, and they have the added benefit of giving insight into your engineering research.
Read math. Get yourself a book and work your way through it in your spare time. Just 1-2 hrs a day, of intense mathematical focus will surprisingly allow you to learn a lot more math than you would think. Start at the right level, for some this will be advanced and for others this will be not so advanced.
think about math a lot. Just ask yourself questions a lot. Math is a mindset. Get away from the computer and coding and work with pen and paper (or a whiteboard) for a couple hours each day. Do problems -- putnum exams, course websites, math.stackexchange. They are all great resources.