4,874 research outputs found
An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is
solved numerically by using the finite difference method in combination with a
convex splitting technique of the energy functional. For the non-stochastic
case, we develop an unconditionally energy stable difference scheme which is
proved to be uniquely solvable. For the stochastic case, by adopting the same
splitting of the energy functional, we construct a similar and uniquely
solvable difference scheme with the discretized stochastic term. The resulted
schemes are nonlinear and solved by Newton iteration. For the long time
simulation, an adaptive time stepping strategy is developed based on both
first- and second-order derivatives of the energy. Numerical experiments are
carried out to verify the energy stability, the efficiency of the adaptive time
stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA
Mathematic
A linear adaptive second-order backward differentiation formulation scheme for the phase field crystal equation
In this paper, we present and analyze a linear fully discrete second order
scheme with variable time steps for the phase field crystal equation. More
precisely, we construct a linear adaptive time stepping scheme based on the
second order backward differentiation formulation (BDF2) and use the Fourier
spectral method for the spatial discretization. The scalar auxiliary variable
approach is employed to deal with the nonlinear term, in which we only adopt a
first order method to approximate the auxiliary variable. This treatment is
extremely important in the derivation of the unconditional energy stability of
the proposed adaptive BDF2 scheme. However, we find for the first time that
this strategy will not affect the second order accuracy of the unknown phase
function by setting the positive constant large enough such
that C_{0}\geq 1/\Dt. The energy stability of the adaptive BDF2 scheme is
established with a mild constraint on the adjacent time step radio
\gamma_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645. Furthermore, a rigorous error
estimate of the second order accuracy of is derived for the proposed
scheme on the nonuniform mesh by using the uniform bound of the
numerical solutions. Finally, some numerical experiments are carried out to
validate the theoretical results and demonstrate the efficiency of the fully
discrete adaptive BDF2 scheme.Comment: 21 pages, 5 figure
An energy-stable time-integrator for phase-field models
We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework
A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation
We propose a novel second order in time numerical scheme for
Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme
is based on second order convex-splitting for the Cahn-Hilliard equation and
pressure-projection for the Navier-Stokes equation. We show that the scheme is
mass-conservative, satisfies a modified energy law and is therefore
unconditionally stable. Moreover, we prove that the scheme is uncondition- ally
uniquely solvable at each time step by exploring the monotonicity associated
with the scheme. Thanks to the weak coupling of the scheme, we design an
efficient Picard iteration procedure to further decouple the computation of
Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by
the mixed finite element method. Ample numerical experiments are performed to
validate the accuracy and efficiency of the numerical scheme
Parallel energy stable phase field simulations of Ni-based alloys system
In this paper, we investigate numerical methods for solving Nickel-based
phase field system related to free energy, including the elastic energy and
logarithmic type functionals. To address the challenge posed by the particular
free energy functional, we propose a semi-implicit scheme based on the discrete
variational derivative method, which is unconditionally energy stable and
maintains the energy dissipation law and the mass conservation law. Due to the
good stability of the semi-implicit scheme, the adaptive time step strategy is
adopted, which can flexibly control the time step according to the dynamic
evolution of the problem. A domain decomposition based, parallel
Newton--Krylov--Schwarz method is introduced to solve the nonlinear algebraic
system constructed by the discretization at each time step. Numerical
experiments show that the proposed algorithm is energy stable with large time
steps, and highly scalable to six thousand processor cores.Comment: arXiv admin note: text overlap with arXiv:2007.0456
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