# Hierarchical Reaction-Diffusion Master Equation

## TL;DR

This paper introduces a hierarchical algorithm for reaction-diffusion simulations that adaptively couples different mesh levels, significantly reducing computational costs while maintaining accuracy in multiscale chemical systems.

## Contribution

A novel multilevel coupling algorithm for reaction-diffusion simulations that efficiently balances accuracy and computational efficiency.

## Key findings

- Achieves up to 1000x speedup compared to microscopic simulations.
- Maintains accuracy in systems where coarse models alone are insufficient.
- Demonstrates effectiveness through multiple numerical examples.

## Abstract

We have developed an algorithm coupling mesoscopic simulations on different levels in a hierarchy of Cartesian meshes. Based on the multiscale nature of the chemical reactions, some molecules in the system will live on a fine-grained mesh, while others live on a coarse-grained mesh. By allowing molecules to transfer from the fine levels to the coarse levels when appropriate, we show that we can save up to three orders of magnitude of computational time compared to microscopic simulations or highly resolved mesoscopic simulations, without losing significant accuracy. We demonstrate this in several numerical examples with systems that cannot be accurately simulated with a coarse-grained mesoscopic model.