WIAS Preprint No. 1004, (2005)

A dissipative discretization scheme for a nonlocal phase segregation model



Authors

  • Gajewski, Herbert
  • Gärtner, Klaus

2010 Mathematics Subject Classification

  • 65M12 65P30 35K65

Keywords

  • Cahn-Hilliard equation, initial boundary value problem, Lyapunov function, stable and unstable steady states, classical thermodynamics, nonlocal phase segregation model

Abstract

We are interested in finite volume discretization schemes and numerical solutions for a nonlocal phase segregation model, suitable for large times and interacting forces. Our main result is a scheme with definite discrete dissipation rate proportional to the square of the driving force for the evolution, i. e., the discrete antigradient of the chemical potential v. Steady states are characterized by constant v and satisfy a nonlocal stationary equation. A numerical bifurcation analysis of that stationary equation explains the observed global behavior of numerically computed trajectories of the evolution equation. For strong interaction forces the model shows steady states distinguished by small deformations of the 'mushy region' or 'interface states'. One essential open question in the discrete case is the global boundedness of v.

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