Discretization of Laplacian Operator on 19 Points Stencil Using Cylindrical Mesh System with the Help of Explicit Finite Difference Scheme

Abstract

Abstract.

In research and technology, discretization is essential for describing and numerically assessing mathematical models. To provide promising numerical formulations with computationally efficient, mathematical models, which are frequently constructed using partial differential equations need to be discretized on curved meshes. Achieving isotropic discrete solutions is crucial for numerical evaluations of partial differential equations in many mathematical models, since it guarantees stability, correctness, and efficiency. This work revolves around the Laplacian operator, a fundamental mathematical operation in models such as Cell Dynamic simulations, Self-Consistent Field theory, and Image Processing. In this work, we utilize the explicit finite difference approach due of its correctness and computational ease. Using a cylindrical mesh structure, the Laplacian is discretized on a 19-point stencil with mixed derivatives. On cylindrical mesh systems, the numerical formulations developed here can be used to approximate solutions of partial differential equations of the first and second order. Significant isotropy, stability, and precision in computational findings for the Laplacian operator are guaranteed by the computational molecule proposed in this work. Potential applications of this formulation include the numerical analysis of different mathematical models employing the Laplacian operator.