arXiv subjects

Hong-lin Liao

Publications and source records attributed to Hong-lin Liao.

35 records · Page 2Linked to original sources

arXiv2021

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 Grönwall inequality were applied to establish an $L^2$ 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.

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arXiv2020

An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense), and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order $α\rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $α\rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative, with the help of a class of discrete orthogonal convolution kernels. This is the first time such a \textit{discrete} transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme.

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arXiv2020

Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators

The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators. More precisely, for a class of discrete convolution kernels relevant to variable-step time discretizations,we show that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Our proof is based on an elementary constructing strategy using the properties of discrete orthogonal convolution kernels and complementary convolution kernels. To the best of our knowledge, this is the first general result on simple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using the unified theory, the stability for some simple non-uniform time-stepping schemes can be obtained in a straightforward way.

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arXiv2020

Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios $r_k:=τ_k/τ_{k-1}<3.561.$ Moreover, with a novel discrete orthogonal convolution kernels argument and some new discrete convolutional inequalities, the $L^2$ norm stability and rigorous error estimates are established, under the same step-ratios constraint that ensuring the energy stability., i.e., $0<r_k<3.561.$ This is known to be the best result in literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

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arXiv2020

An adaptive BDF2 implicit time-stepping method for the phase field crystal model

An adaptive BDF2 implicit time-stepping method is analyzed for the phase field crystal model. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if the time-step ratios $r_k:=τ_k/τ_{k-1}<3.561$, a recent zero-stability restriction of variable-step BDF2 scheme for ordinary differential problems. By using the discrete orthogonal convolution kernels and the corresponding convolution inequalities, an optimal $L^2$ norm error estimate is established under the weak step-ratio restriction $0<r_k<3.561$ ensuring the energy stability. This is the first time such error estimate is theoretically proved for a nonlinear parabolic equation. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is suggested to efficiently capture the multi-scale behaviors and to accelerate the numerical simulations.

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arXiv2020

On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation

In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=τ_k/τ_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

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arXiv2019

Analysis of adaptive BDF2 scheme for diffusion equations

The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios $r_k:=τ_k/τ_{k-1}\le(3+\sqrt{17})/2\approx3.561$, the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the $L^2$ norm. The second-order temporal convergence can be recovered if almost all of time-step ratios $r_k\le 1+\sqrt{2}$ or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the $H^1$ seminorm) and the $L^2$ norm monotonicity at the discrete levels. An example is included to support our analysis.

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arXiv2019

Adaptive linear second-order energy stable schemes for time-fractional Allen-Cahn equation with volume constraint

A time-fractional Allen-Cahn equation with volume constraint is first proposed by introducing a nonlocal time-dependent Lagrange multiplier. Adaptive linear second-order energy stable schemes are developed for the proposed model by combining invariant energy quadratization and scalar auxiliary variable approaches with the recent L1$^{+}$ formula. The new developed methods are proved to be volume-preserving and unconditionally energy stable on arbitrary nonuniform time meshes. The accelerated algorithm and adaptive time strategy are employed in numerical implement. Numerical results show that the proposed algorithms are computationally efficient in multi-scale simulations, and appropriate for accurately resolving the intrinsically initial singularity of solution and for efficiently capturing the fast dynamics away initial time.

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arXiv2019

A second-order and nonuniform time-stepping maximum-principle preserving scheme for time-fractional Allen-Cahn equations

In this work, we present a second-order nonuniform time-stepping scheme for the time-fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete maximum principle, and by using the convolution structure of consistency error, we present sharp maximum-norm error estimates which reflect the temporal regularity. As our analysis is built on nonuniform time steps, we may resolve the intrinsic initial singularity by using the graded meshes. Moreover, we propose an adaptive time-stepping strategy for large time simulations. Numerical experiments are presented to show the effectiveness of the proposed scheme. This seems to be the first second-order maximum principle preserving scheme for the time-fractional Allen-Cahn equation.

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arXiv2019

A fourth-order compact solver for fractional-in-time fourth-order diffusion equations

A fourth-order compact scheme is proposed for a fourth-order subdiffusion equation with the first Dirichlet boundary conditions. The fourth-order problem is firstly reduced into a couple of spatially second-order system and we use an averaged operator to construct a fourth-order spatial approximation. This averaged operator is compact since it involves only two grid points for the derivative boundary conditions. The L1 formula on irregular mesh is considered for the Caputo fractional derivative, so we can resolve the initial singularity of solution by putting more grid points near the initial time. The stability and convergence are established by using three theoretical tools: a complementary discrete convolution kernel, a discrete fractional Gronwall inequality and an error convolution structure. Some numerical experiments are reported to demonstrate the accuracy and efficiency of our method.

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arXiv2019

Simple maximum-principle preserving time-stepping methods for time-fractional Allen-Cahn equation

Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo's derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time-steps are always incorporated into our approaches so that an adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Grönwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.

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arXiv2019

Adaptive second-order Crank-Nicolson time-stepping schemes for time fractional molecular beam epitaxial growth models

Adaptive second-order Crank-Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional Molecular Beam Epitaxial models with Caputo's derivative. Based on the piecewise linear interpolation, the Caputo's fractional derivative is approximated by a novel second-order formula, which is naturally suitable for a general class of nonuniform meshes and essentially preserves the positive semi-definite property of integral kernel. The resulting Crank-Nicolson SAV time-stepping schemes are unconditional energy stable on nonuniform time meshes, and are computationally efficient in multiscale time simulations when combined with adaptive time steps, such as are appropriate for accurately resolving the intrinsically initial singularity of solution and for efficiently capturing fast dynamics away from the initial time. Numerical examples are presented to show the effectiveness of our methods.

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arXiv2019

A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem

Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using linear and quadratic polynomial interpolation. Our analysis relies on two tools: a discrete fractional Grönwall inequality and the global consistency analysis. The new consistency analysis makes use of an interpolation error formula for quadratic polynomials, which leads to a convolution-type bound for the local truncation error. To exploit these two tools, some theoretical properties of the discrete kernels in the numerical Caputo formula are crucial and we investigate them intensively in the nonuniform setting. Taking the initial singularity of the solution into account, we obtain a sharp error estimate on nonuniform time meshes. The fully discrete scheme generates a second-order accurate solution on the graded mesh provided a proper grading parameter is employed. An example is presented to show the sharpness of our analysis.

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arXiv2018

A discrete Grönwall inequality with application to numerical schemes for subdiffusion problems

We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose derivatives are singular at~$t=0$. The main result is a type of fractional Grönwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar L1 approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.

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arXiv2018

Sharp $H^1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems

Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional $H^1$-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accuracy). To recover the theoretical accuracy in time, we propose an improved discrete Grönwall inequality and apply it to the well-known L1 formula and a fractional Crank-Nicolson scheme. With the help of a time-space error-splitting technique and the global consistency analysis, sharp $H^1$-norm error estimates of the two nonuniform approaches are established for a reaction-subdiffusion problems. Numerical experiments are included to confirm the sharpness of our analysis.

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arXiv2018

Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations

A fast two-level linearized scheme with unequal time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations because it significantly reduces the computational cost $O(MN^2)$ and storage $O(MN)$ for the standard L1 formula to $O(MN\log N)$ and $O(M\log N)$, respectively, for $M$ grid points in space and $N$ levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time $t=0$, and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a new discrete fractional Grönwall inequality, a global consistency analysis and a discrete $H^2$ energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.

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arXiv2016

Analysis of $L1$-Galerkin FEMs for time-fractional nonlinear parabolic problems

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of $L1$-Galerkin finite element methods. The analysis of $L1$ methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the $L1$ approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to three examples: linear Fokker-Planck equation, nonlinear Huxley equation and Fisher equation.

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