Lately i've been trying to decide my thesis theme and i've become interested in adaptive finite elements and finite volumes algorithms. However, I need my thesis to fit into a physics related theme. Since up until now I've been working on the Gross-Pitaevskii Equation ( fancy name for the NLSE) it would be excellent to marry both things in my thesis.
I'm aware that, conventionally, to solve the non-linear Schrodinger Equation we employ the Strang Splitting scheme and a pseudo-spectral method, and this is pretty efficient.
Would there be any sort of advantage ( that I could use as an excuse ) to use an adaptive algorithm to solve the NLSE?
I'm mainly interested in studying the quantum vortex dynamics.