Numerical solution for two-dimensional partial differential equations using SM’s method

Abstract

Abstract In this research paper, the authors aim to establish a novel algorithm in the finite difference method (FDM). The novel idea is proposed in the mesh generation process, the process to generate random grids. The FDM over a randomly generated grid enables fast convergence and improves the accuracy of the solution for a given problem; it also enhances the quality of precision by minimizing the error. The FDM involves uniform grids, which are commonly used in solving the partial differential equation (PDE) and the fractional partial differential equation. However, it requires a higher number of iterations to reach convergence. In addition, there is still no definite principle for the discretization of the model to generate the mesh. The newly proposed method, which is the SM method, employed randomly generated grids for mesh generation. This method is compared with the uniform grid method to check the validity and potential in minimizing the computational time and error. The comparative study is conducted for the first time by generating meshes of different cell sizes, i.e. , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mn>10</m:mn><m:mo>×</m:mo><m:mn>10</m:mn><m:mo>,</m:mo><m:mspace width=".25em"/><m:mn>20</m:mn><m:mo>×</m:mo><m:mn>20</m:mn><m:mo>,</m:mo><m:mspace width=".25em"/><m:mn>30</m:mn><m:mo>×</m:mo><m:mn>30</m:mn><m:mo>,</m:mo><m:mspace width=".25em"/><m:mn>40</m:mn><m:mo>×</m:mo><m:mn>40</m:mn></m:math> 10\times 10,\hspace{.25em}20\times 20,\hspace{.25em}30\times 30,\hspace{.25em}40\times 40 using MATLAB and ANSYS programs. The two-dimensional PDEs are solved over uniform and random grids. A significant reduction in the computational time is also noticed. Thus, this method is recommended to be used in solving the PDEs.