This research paper describes a new mesoscopic-level simulator for generating time trajectories of a chemical system governed by diffusion and chemical reactions. A mesoscopic-level simulator achieves good balance between computational efficiency and numerical accuracy by dividing the domain into a number of elements, and limiting reactions to within an element and diffusion to between neighbor elements. An underlying assumption is that the concentration of molecules within an element is homogeneous. When this assumption is violated, the simulation can generate inaccurate trajectories. The proposed method adaptively reduces the mesh size at inhomogeneous areas while keeping the coarse mesh size at homogeneous areas. The authors report a reduction in computation time of about 80 percent, based on experiments with Fisher-Kolmogorov and Gray-Scott models.

The novelty of the work lies in how the mesh is refined adaptively. For each element, the gradient of the concentration is examined in two directions (or three for the 3D domain). If the gradient is larger than the threshold level derived from the homogeneous concentration assumption and Poisson distributions, the element is partitioned into four (or eight for 3D) equal size elements. Upon refinement, the number of molecules in the coarse element is distributed among the four finer elements according to the ratios determined by the concentration of the molecules in the neighbor elements.

Overall, the paper is clear and interesting. It does not require any background in computational chemistry. The technique can be useful for image synthesis applications. However, since its correctness was verified only with a heat equation, more empirical studies are needed before it can be used widely for the analysis of physical systems.