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