Exponential Compact Higher Order Scheme for Nonlinear Steady Convection-Diffusion Equations

This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations(CDE)with variable and nonlinear convection coeffi-cients.The scheme is O(h4)for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm.For twodimensional problems,the scheme produces an O(h4+k4)accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure.The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive.Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods.The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE,where the solutions have the sharp gradient at the solution boundary.

Higher-Order Compact Scheme for the Incompressible Navier-Stokes Equations in Spherical Geometry

A higher-order compact scheme on the nine point 2-D stencil is developed for the steady stream-function vorticity form of the incompressible Navier-Stokes(NS)equations in spherical polar coordinates,which was used earlier only for the cartesian and cylindrical geometries.The steady,incompressible,viscous and axially symmetric flow past a sphere is used as a model problem.The non-linearity in the N-S equations is handled in a comprehensive manner avoiding complications in calculations.The scheme is combined with the multigrid method to enhance the convergence rate.The solutions are obtained over a non-uniform grid generated using the transformation r=ex while maintaining a uniform grid in the computational plane.The superiority of the higher order compact scheme is clearly illustrated in comparison with upwind scheme and defect correction technique at high Reynolds numbers by taking a large domain.This is a pioneering effort,because for the first time,the fourth order accurate solutions for the problem of viscous flow past a sphere are presented here.The drag coefficient and surface pressures are calculated and compared with available experimental and theoretical results.It is observed that these values simulated over coarser grids using the present scheme aremore accuratewhen compared to other conventional schemes.It has also been observed that the flow separation initially occurred at Re=21.