An unconditionally stable finite-difference method for the solution of multi-dimensional transport equation

The rightward representation of the Barakat-Clark ADE scheme is extended for the solution of the Multidimensional Transport equation. The first-order derivative of the Transport equation is represented by a one-sided multi-level finite-difference. The resulting scheme is an explicit marching type iterative solution. The visibility of using this method for the solution of the Multi-dimensional Transport equations is demonstrated through the solution of each of the Burgers? equation and the Graetz-Nusselt problem for the thermally developing flow between two-parallel plates at constant temperature (at high and low Peclet numbers) with parabolic velocity distribution. The results are compared with the solutions using other schemes. All of the obtained results are compared with the exact solutions of the analyzed problems. The results show that the proposed scheme is unconditionally stable, better accuracy, faster convergence, and lower storage capacity requirement. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-ncnd/4.0/).

Automated summary

The rightward representation of the Barakat-Clark ADE scheme is extended for solving the Multidimensional Transport equation, demonstrating its feasibility through solving the Burgers' equation and Graetz-Nusselt problem. The proposed scheme shows unconditional stability, better accuracy, faster convergence, and lower storage capacity, as compared with other schemes, indicating its potential for practical applications in solving complex transport equations.