A finite-difference ghost-point multigrid method for multi-scale modelling of sorption kinetics of a surfactant past an oscillating bubble

Clarissa Astuto, Armando Coco*, Giovanni Russo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


We propose a method for the numerical solution of a multiscale model describing sorption kinetics of a surfactant around an oscillating bubble. The evolution of the particles is governed by a convection-diffusion equation for the surfactant concentration c, with suitable boundary condition on the bubble surface, which models the action of the short range attractive-repulsive potential acting on them when they get sufficiently close to the surface [1]. In the domain occupied by the fluid, the particles are transported by the fluid motion generated by the bubble oscillations. The method adopted to solve the equation for c is based on a finite-difference scheme on a uniform Cartesian grid and implemented in 2D and 3D axisymmetric domains. We use a level-set function to define the region occupied by the bubble, while the boundary conditions are discretized by a ghost-point technique to guarantee second order accuracy at the curved boundary. The sparse linear system is finally solved with a geometric multigrid technique designed ad-hoc for this specific problem. Several accuracy tests are provided to prove second order accuracy in space and time. The fluid dynamics generated by the oscillating bubble is governed by the Stokes equation solved with a second order accurate method based on a monolithic approach, where the momentum and continuity equations are solved simultaneously. Since the amplitude of the bubble oscillations are very small, a simplified model is presented where the computational bubble is actually steady and its oscillations are represented purely with time-dependent boundary conditions. A numerical comparison with the moving domain model confirms that this simplification is perfectly reasonable for the class of problems investigated in this paper.

Original languageEnglish (US)
Article number111880
JournalJournal of Computational Physics
StatePublished - Mar 1 2023

Bibliographical note

Funding Information:
G.R. and C.A. acknowledge partial support from ITN-ETN Horizon 2020 Project ModCompShock, Modeling and Computation on Shocks and Interface (Project Reference 642768 ), and from PRIN Project 2017 entitled “Innovative numerical methods for evolutionary partial differential equations and applications” (No. 2017KKJP4X ), coming from the Italian Ministry of Education, University and Research (MIUR). The work of A.C. has been partially supported by the CINECA ISCRA project (class C): BCMG - HP10C7YOPZ . All authors acknowledge support from GNCS–INDAM (National Group for Scientific Computing, Italy).

Publisher Copyright:
© 2022 Elsevier Inc.


  • Ghost-point
  • Level-set methods
  • Moving domains
  • Multigrid for curved boundaries
  • Small amplitude bubble oscillations
  • Stokes problems

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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