ONE-DAY WORKSHOP ON APPLIED MATHEMATICS 2024

SPEAKER

Armando Coco (Università degli Studi di Catania)

SCHEDULE

12:00 - 13:00

TITLE

Efficient Multigrid Techniques for PDEs on Curved Boundaries: Applications in Sorption Kinetics

ABSTRACT

This talk proposes an efficient approach to improve multigrid (MG) methods in solving PDEs across complex-shaped domains. In literature, a common practice is to add extra relaxations near the boundary, but in cases where a refined mesh near the boundary is essential, this approach becomes suboptimal. We build upon prior work to embed boundary conditions in MG techniques on curved boundaries, using a ghost-point technique and a novel local fictitious time step. The technique, demonstrated on uniform Cartesian grids, serves as a foundation for future MG implementations in adaptive mesh refinement (AMR), offering computational cost benefits.

The method is applied to solve multiscale models describing surfactant sorption kinetics around an oscillating bubble. The particles' evolution is governed by a convection-diffusion equation for surfactant concentration, with specific boundary conditions on the bubble surface reflecting short-range attractive-repulsive potentials. The method employs a finite-difference scheme on a uniform Cartesian grid. Ghost-point techniques ensure second-order accuracy at the curved boundary, while a geometric multigrid technique solves the sparse linear system. Accuracy tests demonstrate second-order accuracy in both space and time.

The fluid dynamics, governed by Stokes equations, are solved with a second-order accurate method in a monolithic approach, solving momentum and continuity equations simultaneously. Numerical comparisons validate the reasonable accuracy of this simplification for the investigated problems.