153 research outputs found
Low regularity integrators for semilinear parabolic equations with maximum bound principles
This paper is concerned with conditionally structure-preserving, low
regularity time integration methods for a class of semilinear parabolic
equations of Allen-Cahn type. Important properties of such equations include
maximum bound principle (MBP) and energy dissipation law; for the former, that
means the absolute value of the solution is pointwisely bounded for all the
time by some constant imposed by appropriate initial and boundary conditions.
The model equation is first discretized in space by the central finite
difference, then by iteratively using Duhamel's formula, first- and
second-order low regularity integrators (LRIs) are constructed for time
discretization of the semi-discrete system. The proposed LRI schemes are proved
to preserve the MBP and the energy stability in the discrete sense.
Furthermore, their temporal error estimates are also successfully derived under
a low regularity requirement that the exact solution of the semi-discrete
problem is only assumed to be continuous in time. Numerical results show that
the proposed LRI schemes are more accurate and have better convergence rates
than classic exponential time differencing schemes, especially when the
interfacial parameter approaches zero.Comment: 24 page
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
A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with general mobility
In this paper, we propose and analyze a linear second-order numerical method
for solving the Allen-Cahn equation with general mobility. The proposed
fully-discrete scheme is carefully constructed based on the combination of
first and second-order backward differentiation formulas with nonuniform time
steps for temporal approximation and the central finite difference for spatial
discretization. The discrete maximum bound principle is proved of the proposed
scheme by using the kernel recombination technique under certain mild
constraints on the time steps and the ratios of adjacent time step sizes.
Furthermore, we rigorously derive the discrete error estimate and
energy stability for the classic constant mobility case and the
error estimate for the general mobility case. Various numerical experiments are
also presented to validate the theoretical results and demonstrate the
performance of the proposed method with a time adaptive strategy.Comment: 25pages, 5 figure
Numerical Analysis of First and Second Order Unconditional Energy Stable Schemes for Nonlocal Cahn-Hilliard and Allen-Cahn Equations
This PhD dissertation concentrates on the numerical analysis of a family of fully discrete, energy stable schemes for nonlocal Cahn-Hilliard and Allen-Cahn type equations, which are integro-partial differential equations (IPDEs). These two IPDEs -- along with the evolution equation from dynamical density functional theory (DDFT), which is a generalization of the nonlocal Cahn-Hilliard equation -- are used to model a variety of physical and biological processes such as crystallization, phase transformations, and tumor growth. This dissertation advances the computational state-of-the-art related to this field in the following main contributions: (I) We propose and analyze a family of two-dimensional unconditionally energy stable schemes for these IPDEs. Specifically, we prove that the schemes are (a) uniquely solvable, independent of time and space step sizes; (b) energy stable, independent of time and space step sizes; and (c) convergent, provided the time step sizes are sufficiently small. (II) We develop a highly efficient solver for schemes we propose. These schemes are semi-implicit and contain nonlinear implicit terms, which makes numerical solutions challenging. To overcome this difficulty, a nearly-optimally efficient nonlinear multigrid method is employed. (III) Via our numerical methods, we are able to simulate crystal nucleation and growth phenomena, with arbitrary crystalline anisotropy, with properly chosen parameters for nonlocal Cahn-Hilliard equation, in a very efficient and straightforward way. To our knowledge these contributions do not exist in any form in any of the previous works in the literature
An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator
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