Regularization of a discrete scheme for a three-dimensional problem of the evolution of the interface of different fluids

A regularized discrete scheme is developed that describes the three-dimensional evolution of the interface between fluids with different viscosities and densities in the Leibenzon–Muskat model. The regularization is achieved by smoothing the kernel of the singular integral involved in the differential equation governing the moving interface. The discrete scheme is tested by solving the problem of a drop of one fluid evolving in a translational flow of another.