11 research outputs found
A M\"untz-Collocation spectral method for weakly singular volterra integral equations
In this paper we propose and analyze a fractional Jacobi-collocation spectral
method for the second kind Volterra integral equations (VIEs) with weakly
singular kernel . First we develop a family of fractional
Jacobi polynomials, along with basic approximation results for some weighted
projection and interpolation operators defined in suitable weighted Sobolev
spaces. Then we construct an efficient fractional Jacobi-collocation spectral
method for the VIEs using the zeros of the new developed fractional Jacobi
polynomial. A detailed convergence analysis is carried out to derive error
estimates of the numerical solution in both - and weighted
-norms. The main novelty of the paper is that the proposed method is
highly efficient for typical solutions that VIEs usually possess. Precisely, it
is proved that the exponential convergence rate can be achieved for solutions
which are smooth after the variable change for a
suitable real number . Finally a series of numerical examples are
presented to demonstrate the efficiency of the method
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
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
Highly efficient schemes for time fractional Allen-Cahn equation using extended SAV approach
In this paper, we propose and analyze high order efficient schemes for the
time fractional Allen-Cahn equation. The proposed schemes are based on the L1
discretization for the time fractional derivative and the extended scalar
auxiliary variable (SAV) approach developed very recently to deal with the
nonlinear terms in the equation. The main contributions of the paper consist
in: 1) constructing first and higher order unconditionally stable schemes for
different mesh types, and proving the unconditional stability of the
constructed schemes for the uniform mesh; 2) carrying out numerical experiments
to verify the efficiency of the schemes and to investigate the coarsening
dynamics governed by the time fractional Allen-Cahn equation. Particularly, the
influence of the fractional order on the coarsening behavior is carefully
examined. Our numerical evidence shows that the proposed schemes are more
robust than the existing methods, and their efficiency is less restricted to
particular forms of the nonlinear potentials
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
Energy regularized models for logarithmic SPDEs and their numerical approximations
Understanding the properties of the stochastic phase field models is crucial
to model processes in several practical applications, such as soft matters and
phase separation in random environments. To describe such random evolution,
this work proposes and studies two mathematical models and their numerical
approximations for parabolic stochastic partial differential equation (SPDE)
with a logarithmic Flory--Huggins energy potential. These multiscale models are
built based on a regularized energy technique and thus avoid possible
singularities of coefficients. According to the large deviation principle, we
show that the limit of the proposed models with small noise naturally recovers
the classical dynamics in deterministic case. Moreover, when the driving noise
is multiplicative, the Stampacchia maximum principle holds which indicates the
robustness of the proposed model. One of the main advantages of the proposed
models is that they can admit the energy evolution law and asymptotically
preserve the Stampacchia maximum bound of the original problem. To numerically
capture these asymptotic behaviors, we investigate the semi-implicit
discretizations for regularized logrithmic SPDEs. Several numerical results are
presented to verify our theoretical findings.Comment: 26 pages, 5 figure
Energy stable and maximum bound principle preserving schemes for the Q-tensor flow of liquid crystals
In this paper, we propose two efficient fully-discrete schemes for Q-tensor
flow of liquid crystals by using the first- and second-order stabilized
exponential scalar auxiliary variable (sESAV) approach in time and the finite
difference method for spatial discretization. The modified discrete energy
dissipation laws are unconditionally satisfied for both two constructed
schemes. A particular feature is that, for two-dimensional (2D) and a kind of
three-dimensional (3D) Q-tensor flows, the unconditional
maximum-bound-principle (MBP) preservation of the constructed first-order
scheme is successfully established, and the proposed second-order scheme
preserves the discrete MBP property with a mild restriction on the time-step
sizes. Furthermore, we rigorously derive the corresponding error estimates for
the fully-discrete second-order schemes by using the built-in stability
results. Finally, various numerical examples validating the theoretical
results, such as the orientation of liquid crystal in 2D and 3D, are presented
for the constructed schemes
Fabrication and Applications of Multi-Fluidic Electrospinning Multi-Structure Hollow and Core–Shell Nanofibers
Recently, electrospinning (ESP) has been widely used as a synthetic technology to prepare nanofibers with unique properties from various raw materials. The applications of functionalized nanofibers have gradually developed into one of the most exciting topics in the field of materials science. In this review, we focus on the preparation of multi-structure fibrous nanomaterials by means of multi-fluidic ESP and review the applications of multi-structure nanofibers in energy, catalysis, and biology. First, the working principle and process of ESP are introduced; then, we demonstrate how the microfluidic concept is combined with the ESP technique to the multi-fluidic ESP technique. Subsequently, the applications of multi-structure nanofibers in energy (Li+/Na+ batteries and Li–S batteries), hetero-catalysis, and biology (drug delivery and tissue engineering) are introduced. Finally, challenges and future directions in this emerging field are summarized
CoS2-TiO2@C Core-Shell fibers as cathode host material for High-Performance Lithium-Sulfur batteries
Owing to the low cost, high energy density, and high theoretical specific capacity, lithium-sulfur batteries have been deemed as a potential choice for future energy storage devices. However, they also have suf-fered from several scientific and technical issues including low conductivity, polysulfides migration, and volume changes. In this study, CoS2-TiO2@carbon core-shell fibers were fabricated through combination of coaxial electrospinning and selective vulcanization method. The core-shell fibers are able to efficiently host sulfur, confine polysulfides, and accelerate intermediates conversion. This electrode delivers an ini-tial specific capacity of 1181.1 mAh g(-1) and a high capacity of 736.5 mAh g(-1) after 300 cycles with high coulombic efficiency over 99.5% (capacity decay of 0.06% per cycle). This strategy of isolating interactant and selective vulcanization provides new ideas for effectively constructing heterostructure materials for lithium-sulfur batteries. (c) 2021 Elsevier Inc. All rights reserved