Deriving appropriate boundary conditions, and accelerating position-jump simulations, of diffusion using non-local...

Area of Science:

• Computational Physics
• Mathematical Modeling
• Physical Chemistry

Background:

• Traditional diffusion models often rely on local jumping, limiting particle movement to nearest neighbors.
• Simulating diffusion processes, especially those with sharp gradients, can be computationally intensive.
• Implementing boundary conditions for non-local processes presents unique challenges.

Purpose of the Study:

• To explore lattice-based position-jump diffusion models incorporating non-local jumping.
• To derive conditions for equivalence between local and non-local jumping models in the continuum limit.
• To develop and validate generalized implementations of Robin and flux boundary conditions for non-local diffusion.

Main Methods:

• Derivation of continuum limit equivalence conditions.
• Generalization of Robin and flux boundary conditions for arbitrary maximum jump lengths.
• Stochastic simulation for validation of boundary condition implementations.
• Development of hybrid local/non-local schemes and biased jumping models.

Main Results:

• Established conditions for equivalence between local and non-local diffusion models.
• Successfully implemented and validated generalized Robin and flux boundary conditions for non-local processes.
• Demonstrated the efficacy of hybrid local/non-local schemes for sharp concentration gradients.
• Showcased substantial time savings in stochastic simulations using non-local jumping.

Conclusions:

• Non-local jumping in lattice-based diffusion models is a viable and efficient alternative to local jumping.
• Generalized boundary conditions enable accurate simulation of non-local diffusion processes.
• Non-local jumping offers significant computational advantages, particularly for complex diffusion scenarios.