A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems

High-order compact exponential finite difference methods for convection-diffusion type problems

https://digitalcommons.odu.edu/cgi/viewcontent.cgi?article=1047&context=mathstat_fac_pubs

Lattice-Boltzmann simulations of the thermally driven 2D square cavity at high Rayleigh numbers

On the basis of the projection method, a higher order compact finite difference algorithm, which possesses a good spatial behavior, is developed for solving the 2D unsteady incompressible Navier-Stokes equations in primitive variable. The present method is established on a staggered grid system and is at least third-order accurate in space. A third-order accurate upwind compact difference approximation is used to discretize the non-linear convective terms, a fourth-order symmetrical compact difference approximation is used to discretize the viscous terms, and a fourth-order compact difference approximation on a cell-centered mesh is used to discretize the first derivatives in the continuity equation. The pressure Poisson equation is approximated using a fourth-order compact difference scheme constructed currently on the nine-point 2D stencil. New fourth-order compact difference schemes for explicit computing of the pressure gradient are also developed on the nine-point 2D stencil. For the assessment of the effectiveness and accuracy of the method, particularly its spatial behavior, a problem with analytical solution and another one with a steep gradient are numerically solved. Finally, steady and unsteady solutions for the lid-driven cavity flow are also used to assess the efficiency of this algorithm. Copyright (C) 2011 John Wiley & Sons, Ltd.

A higher order compact finite difference algorithm for solving the incompressible Navier-Stokes equations

Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation

A fourth-order compact finite difference scheme on the nine-point 2D stencil is formulated for solving the steady-state Navier-Stokes/Boussinesq equations for two-dimensional, incompressible fluid flow and heat transfer using the stream function-vorticity formulation. The main feature of the new fourth-order compact scheme is that it allows point-successive overrelaxation (SOR) or point-successive under-relaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature

Zhenfu Tian

Papers

A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations