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

Area of Science:

• Numerical analysis
• Computational mathematics
• Fractal dynamics

Background:

• Fractal convection-diffusion models present significant numerical challenges.
• Existing methods lack high-resolution schemes for fractional-order systems.
• Anomalous diffusion requires specialized computational approaches.

Purpose of the Study:

• To introduce a high-resolution compact discretization method for nonlinear fractal convection-diffusion models.
• To develop second and fourth-order numerical schemes for 2D fractional-order convection-dominated anomalous diffusion equations.
• To demonstrate the method's applicability to various fractal partial differential equations.

Main Methods:

• Employing the Hausdorff distance metric for fractal geometry.
• Averaging forward and backward mesh stencils for anomalous diffusion estimation.
• Approximating higher-order fractional derivatives on a minimum mesh stencil.
• Developing second and fourth-order compact discretization schemes.

Main Results:

• A computationally efficient, high-resolution method for fractal convection-diffusion models.
• Demonstrated convergence for nonlinear partial differential equations using the Hausdorff fractal distance metric.
• Successful numerical simulations for fractal Graetz-Nusselt, Poisson, Schrödinger, and anomalous diffusion equations.

Conclusions:

• The proposed high-resolution compact discretization method is effective for solving nonlinear fractal partial differential equations.
• The method is computationally efficient, requiring minimal data storage.
• The technique provides a robust framework for analyzing anomalous diffusion and other phenomena in fractal media.