A Lyapunov Formulation for Efficient Solution of the Poisson and Convection-Diffusion Equations by the Differential Quadrature Method

TL;DR

This paper introduces a Lyapunov matrix formulation for the differential quadrature method to efficiently solve Poisson and convection-diffusion equations, significantly reducing computational effort compared to traditional methods.

Contribution

The study develops a Lyapunov algebraic matrix equation approach for DQ formulations, enabling faster solutions and easier extension to three-dimensional problems.

Findings

Lyapunov formulation simplifies DQ equations for Poisson and convection-diffusion problems.

Fast algorithms for Lyapunov equations reduce computational effort.

Method extension to 3D problems is straightforward.

Abstract

Civan and Sliepcevich [1, 2] suggested that special matrix solver should be developed to further reduce the computing effort in applying the differential quadrature (DQ) method for the Poisson and convection-diffusion equations. Therefore, the purpose of the present communication is to introduce and apply the Lyapunov formulation which can be solved much more efficiently than the Gaussian elimination method. Civan and Sliepcevich [2] first presented DQ approximate formulas in polynomial form for partial derivatives in tow-dimensional variable domain. For simplifying formulation effort, Chen et al. [3] proposed the compact matrix form of these DQ approximate formulas. In this study, by using these matrix approximate formulas, the DQ formulations for the Poisson and convection-diffusion equations can be expressed as the Lyapunov algebraic matrix equation. The formulation effort is simplified, and a simple and explicit matrix formulation is obtained. A variety of fast algorithms in the solution of the Lyapunov equation [4-6] can be successfully applied in the DQ analysis of these two-dimensional problems, and, thus, the computing effort can be greatly reduced. Finally, we also point out that the present reduction technique can be easily extended to the three-dimensional cases.