A spectral resolutioned exponential compact higher order scheme (SRECHOS) is developed to solve stationary convection-diffusion type of differential equations with constant convection and diffusion coefficients. The scheme is O (h6) for one-dimensional problems and produces a tri-diagonal system of equations that can be solved efficiently using Thomas algorithm. For two-dimensional 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 (ADI) procedure. The efficiency of the developed scheme is measured using wave number analysis. The analysis shows that the SRECHOS has a much better spectral resolution than any of the existing higher order schemes. Numerical examples are solved and better performance of the SRECHOS in terms of accuracy over the existing schemes in the literature is demonstrated. © 2008 Elsevier B.V. All rights reserved.

Sanyasiraju V.S.S. Yedida

Department of Mathematics

Differential equations

Differentiation (calculus)

Diffusion

Diffusion in liquids

Heat convection

Linear equations

Powders

Semiconductor doping

COMPACT EXPONENTIAL

CONVECTION-DIFFUSION

Finite difference

Higher order

SPECTRAL RESOLUTION

Difference equations

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Computer Methods in Applied Mechanics and Engineering

Exponential compact higher-order schemes have been developed for unsteady convection-diffusion equation (CDE). One of the developed scheme is sixth-order accurate which is conditionally stable for the Péclet number 0 ≤ Pe ≤ 2.8 and the other is fourth-order accurate which is unconditionally stable. Schemes for two-dimensional (2D) problems are made to use alternate direction implicit (ADI) algorithm. Example problems are solved and the numerical solutions are compared with the analytical solutions for each case. © World Scientific Publishing Company.

International Journal of Computational Methods

Exponential compact higher-order schemes and their stability analysis for unsteady convection-diffusion equations

An Exponential Compact Higher Order (ECHO) scheme is developed for the stream function voracity form of steady viscous incompressible Navier-Stokes equations. The coupled equations are solved using an iterative procedure. The developed scheme has been validated initially using two model problems with known analytic solutions and demonstrated the fourth order rate of convergence and about fifth order for highly convection dominated problems. Thus the developed scheme requires smaller number of grid points for the computation. Finally the tested code is used to simulate the lid-driven square cavity flow up to Reynolds number 10,000. © 2012 2012 Taylor and Francis Group LLC.

Engineering Applications of Computational Fluid Mechanics

Exponential compact higher order scheme for steady incompressible navier-stokes equations

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems. © 2011 Published by Elsevier B.V. on behalf of IMACS.

Mathematics and Computers in Simulation

Efficient exponential compact higher order difference scheme for convection dominated problems

This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. 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. © 2011 Global-Science Press.

Communications in Computational Physics

Exponential compact higher order scheme for nonlinear steady convection-diffusion equations

An exponential compact higher order scheme is developed for stationary convection-diffusion type of differential equations which inludes incompressible flow equations in stream function vorticity formulation. The scheme is, ingeneral, fourth order accurate however for one-dimensional constant convection and diffusion coefficients, the scheme is O(h6) and produces a tri-diagonal system of equations that can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an 0(h4+ k6) accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit (ADI) procedure. The efficiency of the developed scheme is measured using wave number analysis. The analysis shows that the developed scheme has a much better spectral resolution than any of the existing higher order schemes. © 2009 American Institute of Physics.

AIP Conference Proceedings

An exponential compact higher order scheme

Journal of Computational Physics

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

A high-order Padé ADI method for unsteady convection–diffusion equations

International Journal for Numerical Methods in Fluids

Fourth-order exponential finite difference methods for boundary value problems of convective diffusion type

Journal of Computational and Applied Mathematics

Ten ways to generate the Il'in and related schemes

Compact finite difference schemes with spectral-like resolution

Spectral resolutioned exponential compact higher order scheme (SRECHOS) for convection-diffusion equations

Journal	Computer Methods in Applied Mechanics and Engineering
ISSN	00457825
Open Access	No