A fourth-order arithmetic average compact finite-difference method for nonlinear singular elliptic PDEs on a 3D smooth quasi-variable grid network

The analysis of nonlinear elliptic PDEs representing stationary convection-dominated diffusion equation, Sine-Gordon equation, Helmholtz equation, and heat exchange diffusion model in a battery often lacks in closed-form solutions. For the long-term behaviour and to assess the quantitative behaviour of the model, numerical treatment is necessary. A novel numerical approach based on arithmetic average compact discretization employing a quasi-variable grid network is proposed for a wide class of nonlinear three-dimensional elliptic PDEs. The method's key benefit is that it applies to singular models and only needs nineteen-point grids with seven functional approximations. Additionally, the suggested method disseminates the truncation error across the domain, which is unrealistic for finite-difference discretization with a fixed step length of grid points. Often, small diffusion anticipates strong oscillation, and tuning the grid stretching parameter helps error dispersion over the domain. The scheme is examined for maximal error bounds and convergence property with the help of a monotone matrix and its irreducible character. The metrics of solution accuracies, mainly root-mean-squared and absolute errors alongside numerical convergence rate, are inspected by different types of variable coefficients, singular and non-singular 3D elliptic PDEs appearing in a convection-diffusion phenomenon. The performance of the numerical solution corroborates the fourth-order convergence on a quasi-variable grid network.