33 research outputs found
Time-fractional Cahn-Hilliard equation: Well-posedness, degeneracy, and numerical solutions
In this paper, we derive the time-fractional Cahn-Hilliard equation from
continuum mixture theory with a modification of Fick's law of diffusion. This
model describes the process of phase separation with nonlocal memory effects.
We analyze the existence, uniqueness, and regularity of weak solutions of the
time-fractional Cahn-Hilliard equation. In this regard, we consider
degenerating mobility functions and free energies of Landau, Flory--Huggins and
double-obstacle type. We apply the Faedo-Galerkin method to the system, derive
energy estimates, and use compactness theorems to pass to the limit in the
discrete form. In order to compensate for the missing chain rule of fractional
derivatives, we prove a fractional chain inequality for semiconvex functions.
The work concludes with numerical simulations and a sensitivity analysis
showing the influence of the fractional power. Here, we consider a convolution
quadrature scheme for the time-fractional component, and use a mixed finite
element method for the space discretization
Inverse asymptotic treatment: capturing discontinuities in fluid flows via equation modification
A major challenge in developing accurate and robust numerical solutions to
multi-physics problems is to correctly model evolving discontinuities in field
quantities, which manifest themselves as interfaces between different phases in
multi-phase flows, or as shock and contact discontinuities in compressible
flows. When a quick response is required to rapidly emerging challenges, the
complexity of bespoke discretization schemes impedes a swift transition from
problem formulation to computation, which is exacerbated by the need to compose
multiple interacting physics. We introduce "inverse asymptotic treatment" (IAT)
as a unified framework for capturing discontinuities in fluid flows that
enables building directly computable models based on off-the-shelf numerics. By
capturing discontinuities through modifications at the level of the governing
equations, IAT can seamlessly handle additional physics and thus enable novice
end users to quickly obtain numerical results for various multi-physics
scenarios. We outline IAT in the context of phase-field modeling of two-phase
incompressible flows, and then demonstrate its generality by showing how
localized artificial diffusivity (LAD) methods for single-phase compressible
flows can be viewed as instances of IAT. Through the real-world example of a
laminar hypersonic compression corner, we illustrate IAT's ability to, within
just a few months, generate a directly computable model whose wall metrics
predictions for sufficiently small corner angles come close to that of NASA's
VULKAN-CFD solver. Finally, we propose a novel LAD approach via
"reverse-engineered" PDE modifications, inspired by total variation diminishing
(TVD) flux limiters, to eliminate the problem-dependent parameter tuning that
plagues traditional LAD. We demonstrate that, when combined with second-order
central differencing, it can robustly and accurately model compressible flows
A novel high-order linearly implicit and energy-stable additive Runge-Kutta methods for gradient flow models
This paper introduces a novel paradigm for constructing linearly implicit and
high-order unconditionally energy-stable schemes for general gradient flows,
utilizing the scalar auxiliary variable (SAV) approach and the additive
Runge-Kutta (ARK) methods. We provide a rigorous proof of energy stability,
unique solvability, and convergence. The proposed schemes generalizes some
recently developed high-order, energy-stable schemes and address their
shortcomings.
On the one other hand, the proposed schemes can incorporate existing SAV-RK
type methods after judiciously selecting the Butcher tables of ARK methods
\cite{sav_li,sav_nlsw}. The order of a SAV-RKPC method can thus be confirmed
theoretically by the order conditions of the corresponding ARK method. Several
new schemes are constructed based on our framework, which perform to be more
stable than existing SAV-RK type methods. On the other hand, the proposed
schemes do not limit to a specific form of the nonlinear part of the free
energy and can achieve high order with fewer intermediate stages compared to
the convex splitting ARK methods \cite{csrk}.
Numerical experiments demonstrate stability and efficiency of proposed
schemes
A linear doubly stabilized Crank-Nicolson scheme for the Allen-Cahn equation with a general mobility
In this paper, a linear second order numerical scheme is developed and
investigated for the Allen-Cahn equation with a general positive mobility. In
particular, our fully discrete scheme is mainly constructed based on the
Crank-Nicolson formula for temporal discretization and the central finite
difference method for spatial approximation, and two extra stabilizing terms
are also introduced for the purpose of improving numerical stability. The
proposed scheme is shown to unconditionally preserve the maximum bound
principle (MBP) under mild restrictions on the stabilization parameters, which
is of practical importance for achieving good accuracy and stability
simultaneously. With the help of uniform boundedness of the numerical solutions
due to MBP, we then successfully derive -norm and -norm
error estimates for the Allen-Cahn equation with a constant and a variable
mobility, respectively. Moreover, the energy stability of the proposed scheme
is also obtained in the sense that the discrete free energy is uniformly
bounded by the one at the initial time plus a {\color{black}constant}. Finally,
some numerical experiments are carried out to verify the theoretical results
and illustrate the performance of the proposed scheme with a time adaptive
strategy