138 research outputs found
Numerical methods for solving the Cahn-Hilliard equation and its applicability to related Energy-based models
In this paper, we review some numerical methods presented in the literature in the last years to approximate the Cahn-Hilliard equation. Our aim is to compare the main properties of each one of the approaches to try to determine which one we should choose depending on which are the crucial aspects when we approximate the equations. Among the properties that we consider desirable to control are the time accuracy order, energy-stability, unique solvability and the linearity or nonlinearity of the resulting systems. In particular, we concern about the iterative methods used to approximate the nonlinear schemes and the constraints that may arise on the physical and computational parameters. Furthermore, we present the connections of the Cahn-Hilliard equation with other physically motivated systems (not only phase field models) and we state how the ideas of efficient numerical schemes in one topic could be extended to other frameworks in a natural way.Ministry of Education, Youth and Sports of the Czech RepublicMinisterio de EconomĂa y Competitivida
Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
Time-fractional partial differential equations are nonlocal in time and show
an innate memory effect. In this work, we propose an augmented energy
functional which includes the history of the solution. Further, we prove the
equivalence of a time-fractional gradient flow problem to an integer-order one
based on our new energy. This equivalence guarantees the dissipating character
of the augmented energy. The state function of the integer-order gradient flow
acts on an extended domain similar to the Caffarelli-Silvestre extension for
the fractional Laplacian. Additionally, we apply a numerical scheme for solving
time-fractional gradient flows, which is based on kernel compressing methods.
We illustrate the behavior of the original and augmented energy in the case of
the Ginzburg-Landau energy functional
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