A transformation‐free HOC scheme for steady convection–diffusion on non‐uniform grids

Abstract

SUMMARYA higher order compact (HOC) ÿnite di erence solution procedure has been proposed for the steady twodimensional (2D) convection-di usion equation on non-uniform orthogonal Cartesian grids involving no transformation from the physical space to the computational space. E ectiveness of the method is seen from the fact that for the ÿrst time, an HOC algorithm on non-uniform grid has been extended to the Navier-Stokes (N-S) equations. Apart from avoiding usual computational complexities associated with conventional transformation techniques, the method produces very accurate solutions for di cult test cases. Besides including the good features of ordinary HOC schemes, the method has the advantage of better scale resolution with smaller number of grid points, with resultant saving of memory and CPU time. Gain in time however may not be proportional to the decrease in the number of grid points as grid non-uniformity imparts asymmetry to some of the associated matrices which otherwise would have been symmetric. The solution procedure is also highly robust as it computes complex ows such as that in the lid-driven square cavity at high Reynolds numbers (Re), for which no HOC results have so far been seen.

High Order Compact Scheme and the Moving Mesh Method for Helmholtz Equation with Variable Wave Numbers

“…Once the step sizes on the both x and y directions are equal, then the mesh is uniform and the scheme (2) has fourth order accuracy. For more details about the derivation process, one can refer to [13,14].…”

“…However, the derivation of the high order compact scheme on nonuniform grid is very nontrivial. The high order compact scheme on nonuniform mesh for 2D convection diffusion equation and 3D Poisson equation are presented in [13,14] and [15], respectively. Numerical results show that the HOC scheme on nonuniform grids is very robust and efficient for the convection diffusion problems with boundary layers and local singularities.…”

Modeling of Shallow-Water Equations by Using Implicit Higher-Order Compact Scheme with Application to Dam-Break Problem

“…Such approximation provides a better representation at shorter length scales and is well applicable for the simulation of waves with high frequency. The application of compact scheme for various fluid flow problems can be obtained from the investigations [1][2][3][4][5][6][7][8][9][10]. Navon and Riphagen [1] developed compact fourth order algorithm by using alternating-direction implicit finite-difference scheme to solve non-linear shallow-water equations, expressed in conservationlaw form.…”

“…Li and Tang [6] extended the previous work of Li et al [11] for steady Navier-Stokes equation to unsteady case by using fourth order accurate compact scheme. Kalita et al [7,9] proposed a transformation-free HOC for steady convectivediffusion equation and an efficient transient Navier-Stokes solver by using non-uniform grids. Pandit et al [8] proposed an implicit HOC finite-difference scheme for two-dimensional unsteady Navier-Stokes equations in irregular geometries by using orthogonal grids.…”

Simulation of flow in two-sided lid-driven square cavities by the lattice Boltzmann method

“…They have found seven and five flow states in antiparallel and parallel motion respectively. Kalita et al [6] developed an HOC algorithm for Stream-function vorticity formulation of the 2D N-S equations on graded Cartesian meshes. They used the algorithm to compute the flow in a two-sided 2D lid-driven cavity [7] where, besides wall shear, free shear flow is also encountered.…”