A new family of implicit fourth order compact schemes for unsteady convection-diffusion equation with variable convection coefficient

TL;DR

This paper introduces a new family of implicit, fourth-order compact finite difference schemes for unsteady convection-diffusion equations with variable convection coefficients, offering high accuracy, stability, and efficiency.

Contribution

The paper presents a novel (5,5) constant coefficient 4th order compact scheme combining compact discretization and Padé approximation, with improved accuracy and stability over standard methods.

Findings

Higher accuracy and better phase/amplitude error characteristics.

Unconditionally stable and capable of using non-unity aspect ratio grids.

Excellent agreement with analytical and numerical results.

Abstract

In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection-diffusion equation with variable convection coefficient is proposed. The schemes are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter. The proposed schemes, where transport variable and its first derivatives are carried as the unknowns, combine virtues of compact discretization and Pad\'{e} scheme for spatial derivative. These schemes which are based on five point stencil with constant coefficients, named as \emph{(5,5) Constant Coefficient 4th Order Compact} [(5,5)CC-4OC], give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods. These schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable. They efficiently capture both transient and steady solutions of linear and nonlinear convection-diffusion equations with Dirichlet as well as Neumann boundary condition. The proposed schemes can be easily implemented and are applied to problems governed by incompressible Navier-Stokes equations apart from linear convection-diffusion equation. Results obtained are in excellent agreement with analytical and available numerical results in all cases, establishing efficiency and accuracy of the proposed scheme.