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A dissipative discretization scheme for a nonlocal phase segregation model
Author(s) -
Gajewski H.,
Gärtner K.
Publication year - 2005
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200510233
Subject(s) - discretization , dissipative system , continuous modelling , bifurcation , dissipation , square (algebra) , mathematics , physics , constant (computer programming) , phase (matter) , statistical physics , classical mechanics , mathematical analysis , nonlinear system , geometry , computer science , thermodynamics , quantum mechanics , programming language
I want to thank H. Gajewski for many years of fruitfulcollaboration and his continuous encouragement, K. Gärtner 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 .