Inexact Picard iterative scheme for steady-state nonlinear diffusion in random heterogeneous media

Area of Science:

• Computational physics
• Materials science
• Applied mathematics

Background:

• Analyzing steady-state nonlinear diffusion in random heterogeneous media presents significant computational challenges.
• Existing methods may struggle with the complexity and stochastic nature of these systems.

Purpose of the Study:

• To develop and present an efficient numerical scheme for analyzing steady-state nonlinear diffusion in random heterogeneous media.
• To validate the proposed scheme against established methods like Monte Carlo simulations.

Main Methods:

• An inexact Picard iteration scheme is employed to solve nonlinear stochastic governing equations.
• Linearization of the nonlinear constitutive law is performed using the current solution estimate.
• Stochastic reduced basis projection schemes are used for spatial discretization and approximation.

Main Results:

• The numerical scheme provides good approximations for response statistics in nonlinear diffusion problems.
• The method demonstrates effectiveness across varying degrees of nonlinearity in a square domain.
• Computational effort is shown to be modest compared to benchmark simulations.

Conclusions:

• The proposed iterative numerical scheme offers an efficient and accurate approach for studying nonlinear diffusion in complex media.
• This method has the potential to advance the analysis of transport phenomena in disordered materials.