ABSTRACT

The numerical solution of Partial Differential Equations (PDEs) by the Finite Difference Method (FDM) is based on the discrete mesh. Since there is no universal mesh generation rule therefore the mesh generation process varies according to the problem and underlying numerical scheme. Various mesh generation techniques for the numerical solution of PDEs have been proposed in literature. But the idea of randomly generated meshes is rarely used. This concept provides the motivation for the investigation of the effects of the randomly generated meshes. Thus, the main objective of this research is to examine the practicability of using the random meshes over uniform meshes. For testing and implementation purpose a 2D Poisson's equation is solved over 100 samples of randomly generated meshes and the feasibility of numerical solution is analyzed by comparing simulation profiles. The effects of the finite difference cell size, average cell size, maximum cell size, and minimum cell size, standard deviation of cell size, skewness of cell size and correlation of cell size are also analyzed. The approximate relationships between the converging iterations over uniform and random meshes are established with 95% significance of regression parameters. These regression models are useful to predict the behavior of random meshes in relation to the uniform meshes. The results are validated by point-wise comparison of the numerical solution obtained over uniform and random meshes and the accuracy of solution is also discussed. Finally, the computational time required to solve the problem over both random and uniform meshes is evaluated. The outcomes of this research have revealed that there is chance that out of 100 samples about 10 to 50 percent random samples of meshes may provide faster convergence using randomly generated meshes as compared to uniform meshes. However, the regularity and smoothness of the solution is violated over random meshes due to unequal and irregular mesh spacing or in other words it can be stated that the high standard deviation in the cell size of random meshes produces high errors in accuracy and convergence of the solution. It is also found that in 30-50% samples of random meshes the computational time is lower than uniform meshes. Further about 77% to 97% numerical solution profiles of random meshes may be closed to the solution of uniform meshes. In summary, the research contributes to the finite difference method using random meshes and helps to decide the practicability and feasibility of using such approaches. It could be concluded that even though the numerical solution over uniform meshes has best properties but there is a chance that about 10 to 50 percent random samples may provide faster convergence than uniform meshes. However, performance of the random meshes may be increased by reshaping the mesh parameters; this work is left for future work with some more directions.