I suppose this is the innovative part. They're not simulating just the string, but also the fluid it's immersed in, which is a computationally hard problem.
I made a vibrating string simulator in college for our Numerical Methods course and for quite a while I couldn't understand why it sounded so bad.
Turns out rounding errors in floating point operations can propagate to a point where they produce this distinct, "metallic" sound.
They're incredibly small, but if your system of differential equations is large enough, they'll become noticeable. Switching to an algorithm with better numerical stability would probably mitigate this issue, but I didn't get that far with my project.