Corrected simulations for one-dimensional diffusion processes with naturally occurring boundaries.

Area of Science:

• Computational Physics
• Stochastic Processes
• Numerical Analysis

Background:

• Simulating diffusion processes often involves discretizing stochastic differential equations using methods like the Euler method.
• Naturally occurring boundaries in diffusion processes pose simulation challenges, as discretized trajectories may cross into forbidden regions.

Purpose of the Study:

• To identify and address the spurious force introduced by the standard Euler method when simulating diffusion with naturally occurring boundaries.
• To develop and validate a corrected numerical scheme that accurately respects diffusion process properties near boundaries.

Main Methods:

• Analysis of the Euler method for one-dimensional diffusion processes with naturally occurring boundaries.
• Development of a corrected numerical scheme to eliminate spurious forces caused by trajectory resetting.
• Numerical testing using exactly soluble one-dimensional diffusion problems with boundaries.

Main Results:

• The standard Euler method with trajectory resetting introduces a significant spurious force, affecting trajectories near boundaries.
• The proposed corrected scheme accurately simulates diffusion processes without introducing spurious forces.
• Numerical tests demonstrate superior accuracy and efficiency of the proposed scheme compared to the standard approach for a given time step.

Conclusions:

• The corrected numerical scheme provides a more accurate and efficient method for simulating one-dimensional diffusion processes with naturally occurring boundaries.
• Eliminating spurious forces is crucial for reliable trajectory simulations in the presence of boundaries.
• The findings have implications for various fields relying on accurate diffusion simulations.