One‐ and two‐step semi‐Lagrangian integrators for arbitrary Lagrangian–Eulerian‐finite element two‐phase flow simulations

Semi-Lagrangian (SL) schemes are of utmost relevance to simulate two-phase flows where advection dominates. The combination of SL schemes with the finite element (FE) method and arbitrary Lagrangian-Eulerian (ALE) dynamic meshes yields a strong ingredient to deal with applications in Earth sciences, environmental engineering, and oil and gas industry whose numerical background is incipient. This article is intended to propose a second-order time multistep SL/ALE/FE scheme to track particle trajectories traveling amidst single- or two-phase incompressible flows both in fixed and moving mesh situations. The novel scheme is a BDF2-like implementation that considers the mesh velocity as part of its backward-in-time integration. Trajectory departure points are searched through extrapolation and followed by a multistep interpolation corrector that works both for constant and varying time steps. A series of two-phase benchmark flow simulations are carried out by using the cubic element to compare the performance of the new integrator against single-step approximations as well as to analyze enhancements in representing the two-phase flow dynamics. We use the 2D axisymmetric Navier-Stokes equations as underlying model. Error analyses, convergence tests, quantitative, and qualitative comparisons are presented and discussed to highlight the superior conservative feature of the novel scheme.