On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations

TL;DR

This paper tackles solving the 2D unsteady nonlinear coupled Burgers' equations in the presence of moderate to severe gradients. It develops and compares two fourth-order schemes—the compact ADI (CADI) scheme and the explicit Du Fort Frankel scheme—employing ADI time stepping and Newton-based linearization. The CADI method is a two-point, fourth-order, A-stable scheme that yields a block-tridiagonal linear system and shows superior stability and efficiency relative to the Du Fort Frankel scheme, as demonstrated by two problem cases. Numerical experiments confirm CADI's ability to deliver accurate solutions on coarser grids and larger time steps, with substantial speedups over Du Fort Frankel and agreement with previous results.

Abstract

The two-dimensional unsteady coupled Burgers' equations with moderate to severe gradients, are solved numerically using higher-order accurate finite difference schemes; namely the fourth-order accurate compact ADI scheme, and the fourth-order accurate Du Fort Frankel scheme. The question of numerical stability and convergence are presented. Comparisons are made between the present schemes in terms of accuracy and computational efficiency for solving problems with severe internal and boundary gradients. The present study shows that the fourth-order compact ADI scheme is stable and efficient.