I am also in academia and I have written quantum mechanical simulation programs to be executed on our cluster. I made the same observation regarding testing or even version control. I was even worse: in my case I am using a C++ library for my simulations and the code I got from others was pure spaghetti code, no inheritance, not even functions.
I rewrote it and I also implemented some unit testing. You are correct that you have to deal with the numerical precision, which can be different depending on the architecture you are running on. Nevertheless, unit testing is possible, as long as you are taking these numerical rounding errors into account. Your result should not depend on the rounding of the numerical values, otherwise you would have a different problem with the robustness of your algorithm.
So, to conclude, I use unit testing for my scientific programs, and it really makes one more confident about the results, especially with regards to publishing the data in the end.