The fluid flow is solved through the finite element discretization of the Navier–Stokes equations, using the semi-Lagrangian method to treat the advective term. A decoupled mesh method based on front-tracking is used to simulate two-phase flows. A coalescence model based on geometric parameters is introduced, and coalescence of circular and Taylor bubbles is presented.

This study explores approximate solutions for fractional Fokker–Planck equations using general finite element schemes developed via the collocation finite element method with trigonometric quintic B-spline basis functions. It validates these methods on two fundamental problems and compares numerical results, including $$$ {L}_2 $$$ and $$$ {L}_{\infty } $$$ error norms across different temporal and spatial steps, against exact solutions and alternative methods. The findings highlight strong agreement between approximate and exact solutions, particularly when employing higher numbers of spatial and temporal elements.

We develop a novel viscosity shuffling Lattice–Boltzmann method to enable the simulation of shear thinning viscoelastic fluids with high viscosity ratios.

A robust and accurate discontinuous Galerkin method is developed for the nonequilibrium multi-material system on tetrahedral meshes. A robust limiting strategy that is crucial for calculations of complex multi-material interactions is presented. A consistency reinstating projection and a shock detector underpin this limiting strategy. Several problems with multi-material interface and shock interactions, like the shock-slug interaction shown here, demonstrate the robustness of the DG method.

Based on three-dimensional seismic wave simulations have become a pivotal aspect of seismic exploration. The diffusive-viscous wave equation, initially proposed by Goloshubin et al., is frequently utilized to describe seismic wave propagation in fluid-saturated media. However, obtaining numerical solutions for this equation has become an urgent issue in recent years. In this study, we present a cell-centered finite volume scheme utilizing a multipoint flux approximation that employs a “diamond stencil” on general polyhedral meshes to address the diffusive-viscous wave equation. Numerical tests exhibit that this new scheme attains optimal convergence, and its effectiveness is demonstrated through simulating vibrations induced by an earthquake source.