80 research outputs found
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 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
Unconditionally Energy Stable Linear Schemes for a Two-Phase Diffuse Interface Model with Peng-Robinson Equation of State
Many problems in the fields of science and engineering, particularly in materials science and fluid dynamic, involve flows with multiple phases and components. From mathematical modeling point of view, it is a challenge to perform numerical simulations of multiphase flows and study interfaces between phases, due to the topological changes, inherent nonlinearities and complexities of dealing with moving interfaces.
In this work, we investigate numerical solutions of a diffuse interface model with Peng-Robinson equation of state. Based on the invariant energy quadratization approach, we develop first and second order time stepping schemes to solve the liquid-gas diffuse interface problems for both pure substances and their mixtures. The resulting temporal semi-discretizations from both schemes lead to linear systems that are symmetric and positive definite at each time step, therefore they can be numerically solved by many efficient linear solvers. The unconditional energy stabilities in the discrete sense are rigorously proven, and various numerical simulations in two and three dimensional spaces are presented to validate the accuracies and stabilities of the proposed linear schemes
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