arXiv subjects

Baoli Yin

Publications and source records attributed to Baoli Yin.

11 recordsLinked to original sources

arXiv2025

A Pick function approach for designing energy-decay preserving schemes of the Maxwell equations in Havriliak-Negami dispersive media

This work proposes a novel approach for designing high-order energy-decaying schemes for Maxwell's equations in Havriliak-Negami dispersive media. It is shown that conventional convolution quadrature (CQ) methods, which rely directly on the generating function of linear multistep methods, cannot generate completely monotonic sequences beyond first-order accuracy. We rigorously prove that for any linear multistep method of second-or higher-order, the associated generating function $\delta(\zeta)$ cannot satisfy both that $-\delta(\zeta)$ is a Pick function and that it is analytic on $(-\infty,1)$ - a key requirement for constructing completely monotonic sequences. To overcome this fundamental limitation, we introduce a reconstruction of the generating function's structure. By strategically incorporating the theory of Pick functions, we successfully construct a second-order completely monotonic sequence. This theoretical advance leads to a discrete scheme that inherits the continuous model's energy decay property, guaranteeing unconditional stability. Numerical experiments confirm the convergence rates and energy dissipation behavior of the proposed method.

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arXiv2025

Analysis of discrete energy-decay preserving schemes for Maxwell's equations in Cole-Cole dispersive medium

This work investigates the design and analysis of energy-decay preserving numerical schemes for Maxwell's equations in a Cole-Cole (C-C) dispersive medium. A continuous energy-decay law is first established for the C-C model through a modified energy functional. Subsequently, a novel $\theta$-scheme is proposed for temporal discretization, which is rigorously proven to preserve a discrete energy dissipation property under the condition $\theta \in [\frac{\alpha}{2}, \frac{1}{2}]$. The temporal convergence rate of the scheme is shown to be first-order for $\theta \neq 0.5$ and second-order for $\theta = 0.5$. Extensive numerical experiments validate the theoretical findings, including convergence tests and energy-decay comparisons. The proposed SFTR-$\theta$ scheme demonstrates superior performance in maintaining monotonic energy decay compared to an alternative 2nd-order fractional backward difference formula, particularly in long-time simulations, highlighting its robustness and physical fidelity.

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arXiv2023

Convolution quadrature for Hadamard fractional calculus and correction methods for the subdiffusion with singular source terms

The convolution quadrature method originally developed for the Riemann-Liouville fractional calculus is extended in this work to the Hadamard fractional calculus by using the exponential type meshes. Local truncation error analysis is presented for singular solutions. By adopting the fractional BDF-$p(1\leq p \leq 6)$ for the Caputo-Hadamard fractional derivative in solving subdiffusion problem with singular source terms, and using the finite element method to discretize the space variable, we carry out the sharp error analysis rigorously and obtain the optimal accuracy by the novel correction technique. Our correction method is a natural generalization of the one developed for subdiffusion problems with smooth source terms. Numerical tests confirm the correctness of our theoretical results.

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arXiv2023

Sharp error analysis for averaging Crank-Nicolson schemes with corrections for subdiffusion with nonsmooth solutions

Thanks to the singularity of the solution of linear subdiffusion problems, most time-stepping methods on uniform meshes can result in $O(τ)$ accuracy where $τ$ denotes the time step. The present work aims to discover the reason why some type of Crank-Nicolson schemes (the averaging Crank-Nicolson scheme) for the subdiffusion can only yield $O(τ^α)$$(α<1)$ accuracy, which is much lower than the desired. The existing well developed error analysis for the subdiffusion, which has been successfully applied to many time-stepping methods such as the fractional BDF-$p (1\leq p\leq 6)$, all requires singular points be out of the path of contour integrals involved. The averaging Crank-Nicolson scheme in this work is quite natural but fails to meet this requirement. By resorting to the residue theorem, some novel sharp error analysis is developed in this study, upon which correction methods are further designed to obtain the optimal $O(τ^2)$ accuracy. All results are verified by numerical tests.

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arXiv2023

A high-order discrete energy decay and maximum-principle preserving scheme for time fractional Allen-Cahn equation

The shifted fractional trapezoidal rule (SFTR) with a special shift is adopted to construct a finite difference scheme for the time-fractional Allen-Cahn (tFAC) equation. Some essential key properties of the weights of SFTR are explored for the first time. Based on these properties, we rigorously demonstrate the discrete energy decay property and maximum-principle preservation for the scheme. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. Comparison with the classic fractional BDF2 and L2-1$_σ$ method further validates the superiority of SFTR in solving the tFAC equation. Experiments concerning both discrete energy decay and discrete maximum-principle also verify the correctness of the theoretical results.

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arXiv2022

Construction of high-order robust theta-methods with applications in anomalous models

A general conversion strategy by involving a shifted parameter $θ$ is proposed to construct high-order accuracy difference formulas for fractional calculus operators. By converting the second-order backward difference formula with such strategy, a novel $θ$-scheme with correction terms is developed for the subdiffusion problem with nonsmooth data, which is robust even for very small $α$ and can resolve the initial singularity.The optimal error estimates are carried out with essential arguments and are verified by numerical tests.

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arXiv2020

Efficient shifted fractional trapezoidal rule for subdiffusion problems with nonsmooth solutions on uniform meshes

This article devotes to developing robust but simple correction techniques and efficient algorithms for a class of second-order time stepping methods, namely the shifted fractional trapezoidal rule (SFTR), for subdiffusion problems to resolve the initial singularity and nonlocality. The stability analysis and sharp error estimates in terms of the smoothness of the initial data and source term are presented. As a byproduct in numerical tests, we find amazingly that the Crank-Nicolson scheme ($θ=\frac{1}{2}$) without initial corrections can restore the optimal convergence rate for the subdiffusion problem with smooth initial data and source terms. To deal with the nonlocality, fast algorithms are considered to reduce the computational cost from $O(N^2)$ to $O(N \log N)$ and save the memory storage from $O(N)$ to $O(\log N)$, where $N$ denotes the number of time levels. Numerical tests are performed to verify the sharpness of the theoretical results and confirm the efficiency and accuracy of initial corrections and the fast algorithms.

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arXiv2020

Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions

We apply two families of novel fractional $θ$-methods, the FBT-$θ$ and FBN-$θ$ methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional $θ$-methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with $O(τ^2+h^{r+1})$ for smooth solutions, where $τ$ is the time mesh size and $h$ is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time.

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arXiv2019

The unified theory of shifted convolution quadrature for fractional calculus

The convolution quadrature theory is a systematic approach to analyse the approximation of the Riemann-Liouville fractional operator $I^α$ at node $x_{n}$. In this paper, we develop the shifted convolution quadrature ($SCQ$) theory which generalizes the theory of convolution quadrature by introducing a shifted parameter $θ$ to cover as many numerical schemes that approximate the operator $I^α$ with an integer convergence rate as possible. The constraint on the parameter $θ$ is discussed in detail and the phenomenon of superconvergence for some schemes is examined from a new perspective. For some technique purposes when analysing the stability or convergence estimates of a method applied to PDEs, we design some novel formulas with desired properties under the framework of the $SCQ$. Finally, we conduct some numerical tests with nonsmooth solutions to further confirm our theory.

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arXiv2018

Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions

In this article, a fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme, which aims at solving nonlinear problems quickly, is considered to numerically solve the nonlinear space fractional Allen-Cahn equations with smooth and non-smooth solutions. The implicit second-order $\theta$ scheme containing both implicit Crank-Nicolson scheme and second-order backward difference method is applied to time direction, a fast TT-M method is used to increase the speed of calculation, and the FE method is developed to approximate the spacial direction. The TT-M FE algorithm includes the following main computing steps: firstly, a nonlinear implicit second-order $\theta$ FE scheme on the time coarse mesh $\tau_c$ is solved by a nonlinear iterative method; secondly, based on the chosen initial iterative value, a linearized FE system on time fine mesh $\tau<\tau_c$ is solved, where some useful coarse numerical solutions are found by the Lagrange's interpolation formula. The analysis for both stability and a priori error estimates are made in detail. Finally, three numerical examples with smooth and non-smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations, from which it is easy to find that the computing time can be saved.

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arXiv2018

TGMFE Algorithm Combined with Some Time Second-Order Schemes for Nonlinear Fourth-Order Reaction Diffusion System

In this article, a two-grid mixed finite element (TGMFE) method with some second-order time discrete schemes is developed for numerically solving nonlinear fourth-order reaction diffusion equation. The two-grid MFE method is used to approximate spatial direction, and some second-order $\theta$ schemes formulated at time $t_{k-\theta}$ are considered to discretize the time direction. TGMFE method covers two main steps: a nonlinear MFE system based on the space coarse grid is solved by the iterative algorithm and a coarse solution is arrived at, then a linearized MFE system with fine grid is considered and a TGMFE solution is obtained. Here, the stability and a priori error estimates in $L^2$-norm for both nonlinear Galerkin MFE system and TGMFE scheme are derived. Finally, some convergence results are computed for both nonlinear Galerkin MFE system and TGMFE scheme to verify our theoretical analysis, which show that the convergence rate of the time second-order $\theta$ scheme including Crank-Nicolson scheme and second-order backward difference scheme is close to $2$, and that with the comparison to the computing time of nonlinear Galerkin MFE method, the CPU-time by using TGMFE method can be saved.

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