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.

A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations.

This paper proposes a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic partial differential equations (PDEs). The underlying new computational method implements the method of so-called approximating fuzzy components, which evaluate the solution values with fourth-order accuracy at internal mesh points. Triangular basic functions and fuzzy components are locally determined by linear combinations of solution values at nine points. Such a scheme connects the proposed method of approximating fuzzy components with the exact values of the solution using a linear system of equations. Compact approximations of high-resolution fuzzy components using nine points give a block tridiagonal Jacobi matrix. Apart from the numerical solution, it is easy to construct closed-form approximate solutions using a 2D spline interpolation polynomial from the available data with fuzzy components. The upper bounds of the approximation errors are estimated, as well as the convergence of the approximating solutions. Simulations with linear and nonlinear elliptical PDEs arising from quantum mechanics and convection-dominated diffusion phenomena are presented to confirm the usefulness of the new scheme and fourth-order convergence. To summarize:•The paper presents a high-resolution numerical method for the two-dimensions elliptic PDEs with nonlinear terms.•The combined effect of fuzzy transform and compact discretizations yields almost fourth-order accuracies to Schro¨dinger equation, convection-diffusion equation, and Burgers equation.•The high-order numerical scheme is computationally efficient and employs minimal data storage.

Method of approximations for the convection-dominated anomalous diffusion equation in a rectangular plate using high-resolution compact discretization.

The present method describes the high-resolution compact discretization method for the numerical solution of the nonlinear fractal convection-diffusion model on a rectangular plate by employing the Hausdorff distance metric. Estimation of anomalous diffusion is formulated by averaging forward and backward mesh stencils. The higher-order fractional derivatives are appropriately approximated on a minimum mesh stencil and subsequently considered for designing a numerical method that falls in the scope of expanded accuracy. Compact discretization is an efficient technique for partial differential equations; however, studies that apply high-resolution scheme for fractional-order systems are still uninvestigated. A second and fourth-order numerical method for the fractional-order convection-dominated anomalous diffusion equation in two dimensions is constructed for practical applications. Convergence of high-order method is obtained for the nonlinear partial differential equations employing Hausdorff fractal distance metric. The numerical simulations with fractal Graetz-Nusselt equation, fractal Poisson equation, fractal Schrödinger equation, and anomalous diffusion equations with variable and constant coefficients are considered to illustrate the utility of the numerical method in the context of local fractional partial differential equations.•The paper demonstrates a computational method for the fractal convection-diffusion model on a rectangular plate.•Two numerical methods of order two and four for the mildly nonlinear fractional-order convection-dominated anomalous diffusion equations are proposed.•The high-resolution scheme is computationally efficient and makes use of minimal data storage.Method name: High-order method for 2D convection-dominated anomalous diffusion equation, Graetz-Nusselt equation, Poisson equation, and Schrödinger equation in fractal media.