66 research outputs found
The variable-step L1 scheme preserving a compatible energy law for time-fractional Allen-Cahn equation
In this work, we revisit the adaptive L1 time-stepping scheme for solving the
time-fractional Allen-Cahn equation in the Caputo's form. The L1 implicit
scheme is shown to preserve a variational energy dissipation law on arbitrary
nonuniform time meshes by using the recent discrete analysis tools, i.e., the
discrete orthogonal convolution kernels and discrete complementary convolution
kernels. Then the discrete embedding techniques and the fractional Gr\"onwall
inequality were applied to establish an norm error estimate on nonuniform
time meshes. An adaptive time-stepping strategy according to the dynamical
feature of the system is presented to capture the multi-scale behaviors and to
improve the computational performance.Comment: 17 pages, 20 figures, 2 table
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
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 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
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