2,439 research outputs found
Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations
Recent results in the literature provide computational evidence that
stabilized semi-implicit time-stepping method can efficiently simulate phase
field problems involving fourth-order nonlinear dif- fusion, with typical
examples like the Cahn-Hilliard equation and the thin film type equation. The
up-to-date theoretical explanation of the numerical stability relies on the
assumption that the deriva- tive of the nonlinear potential function satisfies
a Lipschitz type condition, which in a rigorous sense, implies the boundedness
of the numerical solution. In this work we remove the Lipschitz assumption on
the nonlinearity and prove unconditional energy stability for the stabilized
semi-implicit time-stepping methods. It is shown that the size of stabilization
term depends on the initial energy and the perturba- tion parameter but is
independent of the time step. The corresponding error analysis is also
established under minimal nonlinearity and regularity assumptions
Anomalous diffusion, clustering, and pinch of impurities in plasma edge turbulence
The turbulent transport of impurity particles in plasma edge turbulence is
investigated. The impurities are modeled as a passive fluid advected by the
electric and polarization drifts, while the ambient plasma turbulence is
modeled using the two-dimensional Hasegawa--Wakatani paradigm for resistive
drift-wave turbulence. The features of the turbulent transport of impurities
are investigated by numerical simulations using a novel code that applies
semi-Lagrangian pseudospectral schemes. The diffusive character of the
turbulent transport of ideal impurities is demonstrated by relative-diffusion
analysis of the evolution of impurity puffs. Additional effects appear for
inertial impurities as a consequence of compressibility. First, the density of
inertial impurities is found to correlate with the vorticity of the electric
drift velocity, that is, impurities cluster in vortices of a precise
orientation determined by the charge of the impurity particles. Second, a
radial pinch scaling linearly with the mass--charge ratio of the impurities is
discovered. Theoretical explanation for these observations is obtained by
analysis of the model equations.Comment: This article has been submitted to Physics of Plasmas. After it is
published, it will be found at http://pop.aip.org/pop
Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential
We present and analyze finite difference numerical schemes for the Allen
Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential.
Both the first order and second order accurate temporal algorithms are
considered. In the first order scheme, we treat the nonlinear logarithmic terms
and the surface diffusion term implicitly, and update the linear expansive term
and the mobility explicitly. We provide a theoretical justification that, this
numerical algorithm has a unique solution such that the positivity is always
preserved for the logarithmic arguments. In particular, our analysis reveals a
subtle fact: the singular nature of the logarithmic term around the values of
and 1 prevents the numerical solution reaching these singular values, so
that the numerical scheme is always well-defined as long as the numerical
solution stays similarly bounded at the previous time step. Furthermore, an
unconditional energy stability of the numerical scheme is derived, without any
restriction for the time step size. The unique solvability and the
positivity-preserving property for the second order scheme are proved using
similar ideas, in which the singular nature of the logarithmic term plays an
essential role. For both the first and second order accurate schemes, we are
able to derive an optimal rate convergence analysis, which gives the full order
error estimate. The case with a non-constant mobility is analyzed as well. We
also describe a practical and efficient multigrid solver for the proposed
numerical schemes, and present some numerical results, which demonstrate the
robustness of the numerical schemes
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