For better or worse, Fortran is still a popular language to write clever PDE schemes in, as it maximizes "time to first, fast-enough-running code".

But for anything with a userbase of more than ~15 people, C/C++ are widely preferred.

Julia is starting to pick up steam here. It's a lot easier to write mixed precision algorithms in since the type system is pretty much designed for efficiently writing generic algorithms (and it doesn't hurt that Julia's ODE solvers are SOTA)