Numerical Framework for Pattern-forming Models on Evolving-in-time Surfaces

In this article we describe a numerical framework for a system of coupled reaction-diffusion equations on an evolving-in-time hypersurface Γ. The proposed framework combines the level set methodology for the implicit description of the time dependent Γ, the Eulerian finite element formulation for the numerical treatment of partial differential equations and the flux-corrected transport scheme for the numerical stabilization of arising advective, resp., convective terms. Major advantages of this scheme are that it avoids numerical calculation of curvature, allows interaction of surface-defined partial differential equations with domain-defined partial differential equations through the level set bulk and preserves the positivity of the solution through the algebraic flux correction. The corresponding numerical tests demonstrate the ability of the scheme to deliver an acceptably accurate solution in a reasonably short time.