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Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology. (English) Zbl 1203.65194

Summary: 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.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65L04 Numerical methods for stiff equations
92B05 General biology and biomathematics

Software:

RADAU; RKC
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Full Text: DOI

References:

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