The Spatial Regime Conversion Method

Cameron, Charles G.

doi:10.3390/math13213406

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The Spatial Regime Conversion Method

2025-10-26

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Cameron, Charles G.

Cameron A. Smith

Christian A. Yates *

DOI：10.3390/math13213406

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Abstract

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We present the spatial regime conversion method (SRCM), a novel hybrid modelling framework for simulating reaction-diffusion systems that adaptively combines stochastic discrete and deterministic continuum representations. Extending the regime conversion method (RCM) to spatial settings, the SRCM employs a discrete reaction-diffusion master equation (RDME) representation in regions of low concentration and continuum partial differential equations (PDEs) where concentrations are high, dynamically switching based on local thresholds. This is an advancement over the existing methods in the literature, requiring no fixed spatial interfaces, enabling efficient and accurate simulation of systems in which stochasticity plays a key role but is not required uniformly across the domain. We specify the full mathematical formulation of the SRCM, including conversion reactions, hybrid kinetic rules, and consistent numerical updates. The method is validated across several one-dimensional test systems, including simple diffusion from a region of high concentration, the formation of a morphogen gradient, and the propagation of FKPP travelling waves. The results show that the SRCM captures key stochastic features while offering substantial gains in computational efficiency over fully stochastic models.

Keywords:

stochastic simulation

PDE modelling

hybrid modelling

reaction-diffusion systems

multiscale modelling

spatially extended hybrid simulation

travelling wave

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