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.

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.

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.