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Application of Discontinuous Galerkin Methods for Reaction-Diffusion Systems in Developmental Biology

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Abstract

Nonlinear reaction-diffusion systems which are often employed in mathematical modeling in developmental biology are usually highly stiff in both diffusion and reaction terms. Moreover, they are typically considered on multidimensional complex geometrical domains because of complex shapes of embryos. We overcome these computational challenges by combining discontinuous Galerkin (DG) finite element methods with Strang type symmetrical operator splitting technique, on triangular meshes. This allows us to avoid directly solving a coupled nonlinear system, as is necessary with the standard implicit schemes. Numerical solutions of two reaction-diffusion systems, the well-studied Schnakenberg model, which has been applied to several problems in developmental biology, and a new biologically based system for skeletal pattern formation in the vertebrate limb, are presented to demonstrate effects of various domain geometries on the resulting biological patterns.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556-4618, USA

    Jianfeng Zhu, Yong-Tao Zhang & Mark Alber

  2. Department of Cell Biology and Anatomy, Basic Science Building, New York Medical College, Valhalla, NY, 10595, USA

    Stuart A. Newman

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  1. Jianfeng Zhu
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  2. Yong-Tao Zhang
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  3. Stuart A. Newman
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  4. Mark Alber
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Zhu, J., Zhang, YT., Newman, S.A. et al. Application of Discontinuous Galerkin Methods for Reaction-Diffusion Systems in Developmental Biology. J Sci Comput 40, 391–418 (2009). https://doi.org/10.1007/s10915-008-9218-4

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