When designing and implementing numerical schemes, it is imperative to consider the stability of the applied methods. Prior research has presented different results for the stability of generalized finite-difference methods applied to advection and diffusion equations. In recent years, research has explored a generalized finite-difference approach to the advection-diffusion equation solved on non-rectangular and highly irregular regions using convex, logically rectangular grids. This paper presents a study on the stability of generalized finite difference schemes applied to the numerical solution of the wave equation, solved on clouds of points for highly irregular domains. The stability analysis presented in this work provides significant insights into the proper discretizations needed to obtain stable and satisfactory results. The proposed explicit scheme is conditionally stable, while the implicit scheme is unconditionally stable. Notably, the stability analyses presented in this paper apply to any scheme which is at least second order in space, not just the proposed approach. The proposed scheme offers effective means of numerically solving the wave equation, particularly for highly irregular domains. By demonstrating the stability of the scheme, this study provides a foundation for further research in this area.

Density-driven groundwater flows are described by nonlinear coupled differential equations. Due to its importance in engineering and earth science, several linearizations and semi-linearization schemes for approximating their solution have been proposed. Among the more efficient are the combinations of Newtonian iterations for the spatially discretized system obtained by either scalar homotopy methods, fictitious time methods, or meshless generalized finite difference method, with several implicit methods for the time integration. However, when these methods are used, some parameters need to be determined, in some cases, even manually. To overcome this problem, this paper presents a novel generalized finite differences scheme combined with an adaptive step-size method, which can be applied for solving the governing equations of interest on non-rectangular structured and unstructured grids. The proposed method is tested on the Henry and the Elder problems to verify the accuracy and the stability of the proposed numerical scheme.

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