A Fourier-Based Comparative Analysis of Discrete 3D Laplacians for Isotropy

Abstract

An elliptic partial differential operator, well known as Laplacian operator has diversified applications in fields of science. Its applications comprise numerical analysis, heat flow equations, polymers, image processing, etc. Our work presents the study of three-dimensional discrete Laplacian operators, their formulations in finite difference schemes to analyse their Isotropies and Fourier Stabilities. A few number of three-dimensional Laplacians have been chosen based on different stencil size which are comprised of 7-point, 15-point, 19-point and a family of 27-point. This research study focuses on the mathematical formulation of 3D Laplacian operators, especially to carry out its finite difference for the best stencil identification to be isotropic and to avoid anomalies (artefacts). Using discrete Fourier analysis, we derive the modified wavenumber symbols for each scheme and evaluate their isotropy and dispersion characteristics. The analysis reveals the directional accuracy and stability properties of these Laplacians, which are critical in numerical solutions of elliptic and parabolic partial differential equations (PDEs) in three dimensions. The surface contours plots show a visual comparison of isotropic behaviour of each Laplacian with ideal Laplacian. The 27-point stencils have achieved improved isotropy which can be used in the simulations of 3D heat diffusion equation, cell dynamical model (CDS) and hydrodynamics lattice. Future work includes the derivations of 3D Laplacians for adaptive grids or irregular domains.