Numerical preservation of velocity induced invariant regions for reaction-diffusion systems on evolving surfaces

Frittelli, Massimo, Madzvamuse, Anotida, Sgura, Ivonne and Venkataraman, Chandrasekhar (2018) Numerical preservation of velocity induced invariant regions for reaction-diffusion systems on evolving surfaces. Journal of Scientific Computing. ISSN 0885-7474

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We propose and analyse a finite element method with mass lumping (LESFEM) for the numerical approximation of reaction-diffusion systems (RDSs) on surfaces in R3 that evolve under a given velocity field. A fully-discrete method based on the implicit-explicit (IMEX) Euler time-discretisation is formulated and dilation rates which act as indicators of the surface evolution are introduced. Under the assumption that the mesh preserves the Delaunay regularity under evolution, we prove a sufficient condition, that depends on the dilation rates, for the existence of invariant regions (i) at the spatially discrete level with no restriction on the mesh size and (ii) at the fully-discrete level under a timestep restriction that depends on the kinetics, only. In the specific case of the linear heat equation, we prove a semi- and a fully-discrete maximum principle. For the well-known activator-depleted and Thomas reaction-diffusion models we prove the existence of a family of rectangles in the phase space that are invariant only under specific growth laws. Two numerical examples are provided to computationally demonstrate (i) the discrete maximum principle and optimal convergence for the heat equation on a linearly growing sphere and (ii) the existence of an invariant region for the LESFEM-IMEX Euler discretisation of a RDS on a logistically growing surface.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics
Depositing User: Billy Wichaidit
Date Deposited: 15 May 2018 14:55
Last Modified: 26 Jun 2018 15:08

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Project NameSussex Project NumberFunderFunder Ref
Mathematical Modelling and Analysis of Spatial Patterning on Evolving SurfacesG0872EPSRC-ENGINEERING & PHYSICAL SCIENCES RESEARCH COUNCILEP/J016780/1
Unravelling new mathematics for 3D cell migrationG1438LEVERHULME TRUSTRPG-2014-149