A level-set multigrid technique for nonlinear diffusion in the numerical simulation of marble degradation under chemical pollutants

TL;DR

This paper introduces a level-set multigrid method for efficiently solving nonlinear diffusion equations modeling marble degradation under chemical pollutants, combining accurate boundary handling with fast convergence.

Contribution

It develops a novel multigrid technique integrated with level-set and ghost-cell methods for nonlinear diffusion problems in complex geometries.

Findings

High-quality reconstruction of marble degradation patterns

Enhanced computational efficiency over traditional methods

Effective handling of complex boundary conditions

Abstract

Having in mind the modelling of marble degradation under chemical pollutants, e.g.~the sulfation process, we consider governing nonlinear diffusion equations and their numerical approximation.The space domain of a computation is the pristine marble object. In order to accurately discretize it while maintaining the simplicity of finite difference discretizations, the domain is described using a level-set technique. A uniform Cartesian grid is laid over a box containing the domain, but the solution is defined and updated only in the grid nodes that lie inside the domain, the level-set being employed to select them and to impose accurately the boundary conditions. We use a Crank-Nicolson scheme in time, while for the space variables the discretization is performed by a standard Finite-Difference scheme for grid points inside the domain and by a ghost-cell technique on the ghost points (by using boundary conditions). The solution of the large nonlinear system is obtained by a Newton-Raphson procedure and a tailored multigrid technique is developed for the inner linear solvers. The numerical results, which are very satisfactory in terms of reconstruction quality and of computational efficiency, are presented and discussed at the end of the paper.