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A Narrow Band Numerical Method for a Surface Reaction-Diffusion System Coupled with Surface Motion

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Abstract

Reaction-diffusion equations on surfaces are widely used for modeling various phenomena in biology. This paper presents a novel numerical method for solving a surface reaction-diffusion system coupled with the evolution of the surface. The coupled system has been used to model the growth of hard tumors. A stabilized trace finite element method is used to discretize the reaction-diffusion system on evolving surfaces. The surface motion is computed using a diffusion-generated method for the level-set function, which involves solving a heat equation in each time step followed by a redistance operation. Both the trace finite element space for the reaction-diffusion system and the finite element space for the level-set function are defined in a narrow band region near the surface on a bulk mesh. The method is fully decoupled and allows for easy handling of topology changes. Numerical experiments demonstrate the efficiency of the proposed method for solving this complex problem.

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Notes

  1. We define the mean curvature as the sum of the principal curvatures rather than the arithmetic mean.

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Acknowledgements

We thank Maxim A. Olshanskii for helpful discussions. We are also grateful to the anonymous referees whose comments have served to greatly improve various parts of the paper.

Funding

This work is partially supported by the National Natural Science Foundation of China (11971469 and 12371415) and by Beijing Natural Science Foundation (Grant No. Z240001).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049, Beijing, China

    Song Lu

  2. LSEC, ICMSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China

    Song Lu & Xianmin Xu

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  1. Song Lu
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Correspondence to Xianmin Xu.

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Lu, S., Xu, X. A Narrow Band Numerical Method for a Surface Reaction-Diffusion System Coupled with Surface Motion. J Sci Comput 103, 2 (2025). https://doi.org/10.1007/s10915-024-02772-2

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