Finite difference method on randomly generated non-uniform meshes for poisson equation / Sanaullah Mastoi - Sanaullah , Mastoi (2021)
In this research, a novel method based on randomly generated grids is proposed. This method 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 finite-difference method involving uniform grids is commonly used to solve the 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 employed randomly generated grids for mesh generation. This method is compared with the uniform grids 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 sizes and boundary values. The numerical solutions of partial differential equations and the generalized classification of fractional differential equations are obtained through various approaches, such as exact solutions, analytically, fractional differentiations, and the more generalized form of finite difference method over uniform novel method randomly generated grids. The proposed method is also known as sanaullah mastoi’s method or SM’s method. The new approach is the numerical solution through the finite difference method using randomly generated grids. This study proves that the finite difference method over randomly generated grids found faster convergence iteratively, reduced computational time than uniform grids, and minimize error. A significant reduction in computational time is also noticed. Thus, this method is recommended to be used in solving the partial differential equation. However, SM’s Method’s performance may be increased by reshaping the mesh parameters, and broad scope of research is available.