In this paper, we present a numerical scheme for solving two-phase or free surface flows. Here, the interface/free surface is modelled using the level-set formulation. Besides, the mesh is anisotropic and adapted at each iteration. This adaptation allows us to obtain a precise approximation for the interface/free-surface location. In addition, it enables us to solve the time-discretized fluid equation only on the fluid domain in the case of free-surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier–Stokes equation which is temporally discretized using the method of characteristics and is solved at each time iteration by a first order Lagrange–Galerkin method. The level-set function representing the interface/free surface satisfies an advection equation which is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level-set function (redistancing) as well as the construction of a convenient flow for the level-set advection equation. Finally, some numerical results are presented for both bi-fluid and free-surface problems.
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