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fast high-accuracy compact conservative difference schemes for solving the nonlinear Schrodinger equation

JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 2022年第1期28卷 10-38页

作者： Almushaira, Mustafa Huazhong Univ Sci & Technol Sch Math & Stat Wuhan 430074 Peoples R China

fast high-order compact finite difference schemes are investigated for solving the two-dimensional nonlinear Schrodinger equation with periodic boundary conditions. These schemes are convergent of order s in space, where (s = 4, 6, 8, 10) with second-order temporal accuracy. The discrete conservation laws and convergence of the finite difference schemes are rigorously demonstrated. Thanks to the circulant matrices resulting from spatial discretization, we significantly reduce the computation complexity and storage requirement of the proposed schemes via fast Fourier transform. Numerical examples are presented to show the accuracy and efficiency of these schemes and verify the theoretical analysis. Moreover, we extend the tenth-order scheme to solve the generalized nonlinear Schrodinger equation.

关键词： Nonlinear Schrodinger equation compact finite difference fast algorithm conservation laws

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Fitted schemes for Caputo-Hadamard fractional differential equations

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NUMERICAL algorithmS 2024年第1期97卷 135-164页

作者： Ou, Caixia Cen, Dakang Wang, Zhibo Vong, Seakweng Guangdong Univ Technol Sch Math & Stat Guangzhou 510006 Guangdong Peoples R China Univ Macau Dept Math Macau Peoples R China

In the present paper, the regularity and finite difference methods for Caputo-Hadamard fractional differential equations with initial value singularity are taken into consideration. To overcome the weak singularity and enhance convergence precision, a fitted scheme on nonuniform meshes is applied to such problems. Firstly, based on L-log,L-2-1 sigma approximation, the temporal convergence accuracy of fitted scheme for the sub-diffusion equations is O(N-- min{2r alpha,N-2}), where N denotes the number of time steps, alpha is the fractional order and r is the mesh grading parameter. It is indicated that the performance of the fitted scheme is better than that of the standard L-log,L-2-1 sigma scheme on (i) exponent meshes (i.e., r = 1) and (ii) graded meshes with the optimal choice of the mesh grading. Secondly, a second-order fitted scheme on exponent meshes for the diffusion-wave equations is obtained. Furthermore, for the sake of improving the computational efficiency and demonstrating the effectiveness of the decomposition of the solution, the fast algorithm and further decomposition of the solution for the sub-diffusion equations are investigated. Ultimately, some examples are presented to verify the availability of our theoretical results.

关键词： Caputo-Hadamard fractional differential equations Weak singularity Fitted scheme fast algorithm Nonuniform meshes

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An Efficient Numerical algorithm for the Model Describing the Competition Between Super- and Sub-diffusions Driven by Fractional Brownian Sheet Noise

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JOURNAL OF SCIENTIFIC COMPUTING 2023年第1期96卷 1-24页

作者： Sun, Jing Nie, Daxin Deng, Weihua Lanzhou Univ Sch Math & Stat Gansu Key Lab Appl Math & Complex Syst Lanzhou 730000 Peoples R China

Super- and sub-diffusions are two typical types of anomalous diffusions in the natural world. In this work, we discuss the numerical scheme for the model describing the competition between super- and sub-diffusions driven by fractional Brownian sheet noise. Based on the obtained regularization result of the solution by using the properties of Mittag-Leffler function and the regularized noise by Wong-Zakai approximation, we make full use of the regularity of the solution operators to achieve optimal convergence of the regularized solution. The spectral Galerkin method and the Mittag-Leffler Euler integrator are respectively used to deal with the space and time operators. In particular, by contour integral, the fast evaluation of the Mittag-Leffler Euler integrator is realized. We provide complete error analyses, which are verified by the numerical experiments.

关键词： Model for anomalous diffusion Fractional Brownian sheet Wong-Zakai approximation Mittag-Leffler Euler integrator Spectral Galerkin method fast algorithm Error analyses

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fast algorithms for integral formulations of steady-state radiative transfer equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2019年 380卷 191-211页

作者： Fan, Yuwei An, Jing Ying, Lexing Stanford Univ Dept Math Stanford CA 94305 USA Stanford Univ Inst Computat & Math Engn Stanford CA 94305 USA

We investigate integral formulations and fast algorithms for the steady-state radiative transfer equation with isotropic and anisotropic scattering. When the scattering term is a smooth convolution on the unit sphere, a model reduction step in the angular domain using the Fourier transformation in 2D and the spherical harmonic transformation in 3D significantly reduces the number of degrees of freedoms. The resulting Fourier coefficients or spherical harmonic coefficients satisfy a Fredholm integral equation of the second kind. We study the uniqueness of the equation and proved an a priori estimate. For a homogeneous medium, the integral equation can be solved efficiently using the FFT and iterative methods. For an inhomogeneous medium, the recursive skeletonization factorization method is applied instead. Numerical simulations demonstrate the efficiency of the proposed algorithms in both homogeneous and inhomogeneous cases and for both transport and diffusion regimes. (C) 2018 Elsevier Inc. All rights reserved.

关键词： fast algorithm Radiative transfer Fredholm integral equation Recursive skeletonization FFT Anisotropic scattering

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Sonography in Pre-clinical Care

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DEUTSCHE MEDIZINISCHE WOCHENSCHRIFT 2024年第15期149卷 912-924页

作者： Knott, Dominik Siegl, Lutz Siegl, Katharina Bundeswehrkrankenhauses Berlin Zent Interdisziplinaren Notaufnahme Berlin Germany Bundeswehrkrankenhaus Zentrums Notfall & Rettungsmed Berlin Germany Bundeswehrkrankenhaus Klin Anasthesiol Intens Med Schmerztherapie & Notf Berlin Germany

"Have you had a sonogram yet?" is a question that is asked several times a day in clinical emergency medicine. Sonography is a quick and non-invasive way of looking into a person. It is now not only a valuable diagnostic tool, but in times of a lack of resources it is also a way of sensibly controlling and accelerating treatment paths for patients.

关键词： Notfallmedizin Point-of-Care-Ultraschall Notfallsonografie fast-algorithmus Polytrauma Reanimation emergency medicine point-of-care ultrasound emergency sonography fast algorithm polytrauma resuscitation

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Sonography in the preclinical setting

NOTARZT

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NOTARZT 2024年第1期40卷 31-43页

作者： Knott, Dominik Siegl, Lutz Siegl, Katharina Bundeswehr Hosp Berlin Gen Med Responsible Training Cent interdisciplinary emerge Berlin Germany Bundeswehr Hosp Berlin Ctr Emergency Rescue Med Berlin Germany Bundeswehr Hosp Berlin Clin Anesthesiol Intens Care Med Pain Therapy & Em Berlin Germany Bundeswehrkrankenhaus Berlin Klin X Scharnhorststr 13 D-10557 Berlin Germany

Preclinical ultrasound is another valuable tool in the repertoire of emergency medical and rescue service personnel. If the question is clear, it can quickly narrow down differential diagnoses and guide further treatment as well as optimise the choice and preparation of the target hospital. With the devices currently available, it is possible to generate technically high-quality ultrasound images, even if image storage assigned to patients is still a challenge. Thanks to the wide availability of sonography, learning formats and expertise, it is possible to achieve a good level of competence in this procedure. Actually, the biggest obstacles are remembering to use it and incorporating it sensibly into the operational procedure, as it is often necessary to modify the approach that has been established for years. Before introducing prehospital ultrasound, staff should be trained and procedures should be jointly defined to make ultrasound a valuable addition to prehospital treatment and patient disposition.

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A structure preserving difference scheme with fast algorithms for high dimensional nonlinear space-fractional Schrodinger equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2021年 425卷 109869-109869页

作者： Yin, Baoli Wang, Jinfeng Liu, Yang Li, Hong Inner Mongolia Univ Sch Math Sci Hohhot 010021 Peoples R China Inner Mongolia Univ Finance & Econ Sch Stat & Math Hohhot 010070 Peoples R China

We aim at analyzing a novel structure preserving difference scheme for the high dimensional nonlinear space-fractional Schrodinger equation. The temporal direction is discretized by the modified Crank-Nicolson scheme, while the spacial variable is approximated by formulas from the shifted convolution quadrature. The energy and mass preserving properties are proved rigorously for the scheme based on the newly developed properties of the coefficients matrix. Further, the optimal convergence rate O(h(2) + tau(2)) is derived in detail where h, tau denote the spacial and temporal mesh sizes, respectively. To save the computing cost, we consider the preconditioned fast BiCG (PF-BiCG) algorithm for the resulting complex systems, with the computing complexity of O(N log N) at each time step and the memory requirement of O(N). Numerical experiments are conducted to confirm our theoretical conclusions and the efficiency of the fast algorithm. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Structure preserving scheme Nonlinear space-fractional Schrodinger equation fast algorithm The shifted convolution quadrature

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Convergence and superconvergence analysis of finite element methods for nonlinear Ginzburg-Landau equation with Caputo derivative

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COMPUTATIONAL & APPLIED MATHEMATICS 2023年第6期42卷 1-32页

作者： Chen, Fang Li, Meng Zhao, Yanmin Tang, Yifa Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China Xuchang Univ Sch Sci Henan Joint Int Res Lab High Performance Computat Xuchang 461000 Peoples R China Chinese Acad Sci Acad Math & Syst Sci LSEC ICMSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China

In this paper, we study and analyze the time fractional Ginzburg-Landau equation (FGLE) using finite element methods (FEMs) in space and L1 scheme in time. The unconditional optimal L-2-norm error estimates are obtained based on the time-space error splitting technique. Using the relation between interpolation operator and Ritz projection operator, we obtain the superclose results in H-1-norm. Furthermore, the global superconvergence results are established through the interpolation postprocessing technique. To overcome the weak singularity of the solution at the initial time and improve the computational efficiency, we adopt the nonuniform L1 scheme in the time direction and built corresponding fast algorithm based on sum-of-exponential technique. Finally, we provide several numerical experiments to verify the theoretical results and demonstrate the advantages of the fast algorithm.

关键词： Nonlinear time FGLE Convergence and superconvergence FEM Time-space error splitting technique fast algorithm

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A fast stable accurate artificial boundary condition for the linearized Green-Naghdi system

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NUMERICAL algorithmS 2022年第4期90卷 1437-1463页

作者： Pang, Gang Ji, Songsong Beihang Univ Sch Math Sci Beijing 102206 Peoples R China Peking Univ Coll Engn HEDPS CAPT & LTCS Beijing 100871 Peoples R China

In this paper, we use a fully discrete Crank-Nicolson scheme to solve the Cauchy problem for one-dimensional linearized Green-Naghdi system with fast convolution boundary condition which is derived through the Pade approximation for the square root function. We also introduce a constant damping term to guarantee the stability. While the damping term meets certain conditions, the stability is proved for the numerical solutions with the fast convolution boundary condition. Therefore, the difficulty of numerical instability which rises in Kazakova and Noble (SIAM J. Num. Anal. 1, 657-683, 2020) is overcome. The computational cost of the convolution integral is reduced from O(N-2) to O(N ln(N)) for the total number of time steps N. Numerical examples verify the theoretical results and demonstrate the performance for the fast numerical method.

关键词： Green-Naghdi system Pade approximation Convolution quadrature Artificial boundary condition fast algorithm Stability

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fast algorithm on Parameter Estimation of Wideband LFM Signal Based on Down-chirp and CS

Fast Algorithm on Parameter Estimation of Wideband LFM Signa...

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IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC)

作者： Wang Kang Ye Wei Lao Guochao Wang Yong Equipment Acad Beijing Peoples R China

ISBN: (纸本)9781479952748

Compressed Sensing (CS) has been successfully applied to the parameter estimation of Linear Frequency Modulation (LFM) signal. Compared to the Nyquist sampling method, far less samples are needed to estimate the frequency parameter. However, the super-resolution estimation of frequency parameter can greatly increase the number of atoms in the over-complete dictionary and it will brings a huge amount of computation. This paper proposes resolutions to this problem. Taking the feature of CS into account that the sampling and compression are completed at the same time, we structure a measurement matrix which can complete the compressive sampling and down-chirp simultaneously. Furthermore, we propose a down-chirp based method and improve it with fast computing strategy to solve the above problem. Simulation results have proved that the frequency parameter can be accurate estimated under a low SNR and sampling condition. Meanwhile, compared with the proposed method, the improved algorithm greatly reduces the scale of over-complete dictionary and the amount of computation, and the estimation time has been cut down significantly.

关键词： Compressed Sensing parameter estimation LFM fast algorithm down-chirp

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