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

Area of Science:

• Computational fluid dynamics
• Multiphysics simulation
• Biomembranes and transport

Background:

• Fluid-solute-structure interaction presents significant computational challenges.
• Accurate modeling of membrane permeability and charged solute transport is crucial.
• Existing numerical methods often struggle with large timesteps and resolving steep gradients.

Purpose of the Study:

• To develop an efficient and stable numerical method for fluid-solute-structure interaction.
• To accurately simulate transport phenomena across membranes, including charged species.
• To enable larger timesteps and local mesh refinement for complex simulations.

Main Methods:

• An immersed boundary method is employed for fluid-solute-structure interaction.
• Linearly implicit timestepping allows for substantially larger stable timesteps.
• Local mesh refinement resolves steep gradients in space charge layers and chemical potential.

Main Results:

• The numerical scheme demonstrates stability with large timesteps compared to explicit methods.
• Accurate resolution of steep gradients associated with space charge layers and chemical potential is achieved.
• The method effectively models membrane permeability control via chemical potential.

Conclusions:

• The developed immersed boundary method offers a robust approach for fluid-solute-structure interaction problems.
• The methodology shows promise for simulating complex transport phenomena in biological and engineered systems.
• Numerical examples confirm the method's capabilities and convergence properties.