A finite element implementation of the incompressible Schrödinger flow method

First proposed by E. Madelung in 1926, the analogy between quantum mechanics and hydrodynamics has been known for a long time; however, its potentialities and the possibility of using the characteristic equations of quantum mechanics to simulate the behaviour of inviscid fluids have not been thoroughly investigated in the past. In this methodology, the incompressible Euler equations are thus substituted by the Schrödinger equation, turning a quasi-linear Partial Differential Equation into a linear one, an algorithm known in the literature as Incompressible Schrödinger Flow. Previous works on the subject used the Fast Fourier Transform method to solve this problem, obtaining promising results, especially in predicting vortex dynamics; this paper aims to implement this novel approach into a Finite Element framework to find a more general formulation better suited for future application on complex geometries and on test cases closer to real-world applications. Simple case studies are presented in this work to analyse the potentialities of this method: the results obtained confirm that this method could potentially have some advantages over traditional Computational Fluid Dynamics method, especially for what concerns computational savings related to the required time discretisation, whilst also introducing new aspects of the algorithm, mainly related to boundary conditions, not addressed in previous works.