Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30127
A Study of Numerical Reaction-Diffusion Systems on Closed Surfaces

Authors: Mei-Hsiu Chi, Jyh-Yang Wu, Sheng-Gwo Chen

Abstract:

The diffusion-reaction equations are important Partial Differential Equations in mathematical biology, material science, physics, and so on. However, finding efficient numerical methods for diffusion-reaction systems on curved surfaces is still an important and difficult problem. The purpose of this paper is to present a convergent geometric method for solving the reaction-diffusion equations on closed surfaces by an O(r)-LTL configuration method. The O(r)-LTL configuration method combining the local tangential lifting technique and configuration equations is an effective method to estimate differential quantities on curved surfaces. Since estimating the Laplace-Beltrami operator is an important task for solving the reaction-diffusion equations on surfaces, we use the local tangential lifting method and a generalized finite difference method to approximate the Laplace-Beltrami operators and we solve this reaction-diffusion system on closed surfaces. Our method is not only conceptually simple, but also easy to implement.

Keywords: Close surfaces, high-order approach, numerical solutions, reaction-diffusion systems.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1127162

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 870

References:


[1] M. Bergdorf, Ivo F. Sbalzarini, and P. Koumoutsakos, “A Lagrangian particle method for reaction–diffusion systems on deforming surfaces”, J. Math. Biol., vol. 61, 2010, pp. 649–663.
[2] S.-G. Chen, M.-H. Chi, and J.-Y. Wu, “High-Order Algorithms for Laplace–Beltrami Operators and Geometric Invariants over Curved Surfaces”, vol 65, 2015, pp.839-865.
[3] S.-G. Chen and J.-Y. Wu, "Estimating normal vectors and curvatures by centroid weights", Comput. Aided Geom. Design, vol. 21, 2004, pp. 447–458.
[4] M. DoCarmo, “Differential geometry of curves and surfaces, Prentice-Hall, Lodon, 1976.
[5] M. DoCarmo, “Riemannian geometry”, Birkhauser, Boston, 1992.
[6] E. J. Fuselier and G. B. Wright, “A high-order kernel method for diffusion and reaction-diffusion equations on surfaces”, J. Sci. Comput., vol. 56, issue 3, 2013, pp. 535-565.
[7] C. Landsberg and A. Voigt, “A multigrid finite element method for reaction-diffusion systems on surfaces”, Comput Visual Sci, vol. 13, 2010, pp. 177–185.
[8] A. Madzvamuse, A. J. Wathen, and P. K. Maini, "A moving grid finite element method applied to a model biological pattern generator", J. Comput. Phys. vol. 190, no. 2, 2003, pp. 478–500.
[9] DW Thompson, “On growth and form”, 2nd end, Cambridge University Press, 1942.
[10] N. Tuncera, A. Madzvamuseb, and A. J. Meir, “Projected finite elements for reaction–diffusion systems on stationary closed surfaces”, Applied Numerical Mathematics, vol. 96, 2015, pp. 45–71.