Finite-volume scheme for a degenerate cross-diffusion model motivated from ion transport

Area of Science:

• Computational Biology
• Mathematical Modeling
• Biophysics

Background:

• Ion transport through biological membranes is crucial for cellular function.
• Modeling ion transport involves complex, degenerate cross-diffusion systems.
• Existing numerical methods face mathematical difficulties due to system degeneracies.

Purpose of the Study:

• To propose an implicit Euler finite-volume scheme for a degenerate cross-diffusion system.
• To address mathematical challenges posed by nonstandard degeneracies in ion transport models.
• To demonstrate the scheme's efficiency and accuracy in simulating ion channels.

Main Methods:

• Developed an implicit Euler finite-volume scheme using two-point flux approximations.
• Employed 'double' upwind mobilities to handle drift terms and electric potential coupling.
• Proved the existence of solutions and established a new discrete Aubin-Lions lemma.

Main Results:

• The scheme's solutions are proven to exist for the fully discrete system.
• The scheme preserves essential properties like nonnegativity and entropy dissipation.
• Numerical simulations of a calcium channel demonstrate the scheme's efficiency.

Conclusions:

• The proposed finite-volume scheme effectively overcomes mathematical degeneracies in ion transport models.
• The method is efficient and accurate for simulating ion transport, even in complex scenarios.
• This work provides a robust computational tool for studying biological membrane phenomena.