Decoupled, Linear, and Energy Stable Finite Element Method for the Cahn-Hilliard-Navier-Stokes-Darcy Phase Field Model

In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn—Hilliard—Navier—Stokes equations in the free flow region and Cahn—Hilliard—Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions. © 2018 Society for Industrial and Applied Mathematics

Existence of Weak Solutions for Nonlinear Time-Fractional p

The existence of weak solution for p-Laplace problem is studied in the paper. By exploiting the relationship between the Nehari manifold and fibering maps and combining the compact imbedding theorem and the behavior of Palais-Smale sequences in the Nehari manifold, the existence of weak solutions is established. By means of the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations of the p-Laplace problem are obtained

A Galerkin Finite Element Method for Numerical Solutions of the Modified Regularized Long Wave Equation

A Galerkin method for a modified regularized long wave equation is studied using finite elements in space, the Crank-Nicolson scheme, and the Runge-Kutta scheme in time. In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. A Fourier stability analysis for the method is shown to be marginally stable. Three invariants of motion are investigated. Numerical experiments are presented to check the theoretical study of this method

Numerical Simulation of Hot Accretion Flow around Bondi Radius

Previous numerical simulations have shown that strong winds can be produced in the hot accretion flows around black holes. Most of those studies focus only on the region close to the central black hole, therefore it is unclear whether the wind production stops at large radii around Bondi radius. Bu et al. 2016 studied the hot accretion flow around the Bondi radius in the presence of nuclear star gravity. They find that when the nuclear stars gravity is important/comparable to the black hole gravity, winds can not be produced around the Bondi radius. However, for some galaxies, the nuclear stars gravity around Bondi radius may not be strong. In this case, whether winds can be produced around Bondi radius is not clear. We study the hot accretion flow around Bondi radius with and without thermal conduction by performing hydrodynamical simulations. We use the virtual particles trajectory method to study whether winds exist based on the simulation data. Our numerical results show that in the absence of nuclear stars gravity, winds can be produced around Bondi radius, which causes the mass inflow rate decreasing inwards. We confirm the results of Yuan et al. which indicates this is due to the mass loss of gas via wind rather convectional motions.Comment: 15 pages, 8 figures, accepted for publication in Ap