One‐ and two‐step semi‐Lagrangian integrators for ALE‐FE 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.