Pressure‐correction projection methods for the time‐dependent micropolar fluids

Abstract

The unsteady micropolar Navier–Stokes equations (MNSE) is a system which describes the evolution of an incompressible fluid whose material particles possess both translational and rotational degrees of freedom. In this article, both the first order and second order pressure‐correction (PC) projection method for the MNSE are proposed. The unconditionally stability analysis corresponding to the temporal semidiscrete version and the fully discrete version of the PC projection method are proved, and the first order error estimation for the temporal semidiscrete version is deduced. Some numerical simulation results are presented to show the effect of PC projection method.

Local discontinuous Galerkin method coupled with the implicit‐explicit Runge–Kutta method for the time‐dependent micropolar fluid equations

“…. Now, let present the fully discrete method based on the combination of the temporal implicit-explicit RK evolution and spatial LDG approximation for the DAEs (13). Suppose Z h is given, and Z i h , i = 1, 2, … , n is obtained, then find Z n+1 h from First-order method:…”

“…For example, the least-squares method, 5 the stabilized finite element method, 6,7 the pressure stabilization method, 8 the penalty method, [9][10][11] and the artificial compressibility method 12 are proposed to overcome LBB constraint. In terms of nonlinearity and multiphysics coupling, Maimaiti et al 13 proposed pressure-correction projection methods for the time-dependent linear micropolar fluid equations. Numerical analysis and comparison of four stabilized finite element methods for the steady micropolar fluid equations was given in Reference 14.…”

Research on three kinds of splitting finite element schemes for 2D/3D unsteady incompressible thermomicropolar fluid equations

“…Similar to the standard pressure correction method, 14‐18,20,21 Step $$$ {2}^{\prime } $$$ in order to solve (21), there are need to supply the Neumann boundary condition of pressure. Which will lead to numerical boundary layers for pressure field, and the loss of accuracy.…”

“…The advantage of the first kind of method is that there is no need to introduce additional artificial boundary conditions, and the consistency with the original saddle-point problem is well maintained, but the disadvantage is that the calculation scale is large and the velocity is slow. The second class of methods solves the decoupling system of velocity and pressure, such as scalar auxiliary variable method, 13 the pressure-corrected projection methods [14][15][16][17][18][19][20][21][22] and the velocity-corrected projection methods, [23][24][25][26] Gauge-Uzawa method, 27,28 operator-splitting method 29,30 and so forth. There kinds of methods are usually faster to calculate, but the disadvantage is that it introduces additional artificial boundary conditions in the decomposition process, which makes it difficult to ensure consistency with the original problem, which may lead to loss of accuracy.…”

A Pressure Correction Projection Finite Element Method for The 2D/3D Time-Dependent Thermomicropolar Fluid Problem

“…In order to avoid solving the saddle-point problem, a common strategy is to use the projection scheme for decoupling calculation [4,5].…”