Numerical Framework for Pattern-Forming Models on Evolving-in-Time Surfaces
AbstractIn this contribution we describe a numerical framework for a system of coupledreaction-diffusion equations В on an evolving-in-time hypersurface$\Gamma$. Numerical tests are empolyed for Turing-type instability on stationary and evolving surfaces.В The proposed framework combines the level set methodology forthe implicit description of the time dependent Р“ , the Eulerianfinite element formulation for the numerical treatment of partialdifferential equations, and the flux-corrected transport scheme for thenumerical stabilization of arising adjective, resp., convective terms.Major advantages of this scheme are that it avoids numerical calculation ofcurvature, allows coupling of surface-defined partialdifferential equations with domain-defined partialdifferential equations through the level set bulk and preserves the positivity of the solution throughthe algebraic flux correction. Corresponding numerical tests demonstratethe ability of the scheme to deliver В highly accurate solutions with a reasonably good convergence behavior.
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