The immersed boundary method for advection-electrodiffusion with implicit timestepping and local mesh refinement

  • Authors:

  • Pilhwa Lee;Boyce E. Griffith;Charles S. Peskin

  • Affiliations:

  • Department of Cell Biology, University of Connecticut Health Center, 263 Farmington Avenue, Farmington, CT 06030-3505, United States;Leon H. Charney Division of Cardiology, Department of Medicine, NYU School of Medicine, 522 First Avenue, New York, NY 10016, United States;Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

We describe an immersed boundary method for problems of fluid-solute-structure interaction. The numerical scheme employs linearly implicit timestepping, allowing for the stable use of timesteps that are substantially larger than those permitted by an explicit method, and local mesh refinement, making it feasible to resolve the steep gradients associated with the space charge layers as well as the chemical potential, which is used in our formulation to control the permeability of the membrane to the (possibly charged) solute. Low Reynolds number fluid dynamics are described by the time-dependent incompressible Stokes equations, which are solved by a cell-centered approximate projection method. The dynamics of the chemical species are governed by the advection-electrodiffusion equations, and our semi-implicit treatment of these equations results in a linear system which we solve by GMRES preconditioned via a fast adaptive composite-grid (FAC) solver. Numerical examples demonstrate the capabilities of this methodology, as well as its convergence properties.