Computational uncertainty and optimal grid size and time step of the Lax-Friedrichs scheme for the 1D advection equation

本文对于应用Lax-Friedrichs格式数值求解一维平流方程,研究数值求解过程中产生的截断误差与舍入误差,以及两种误差逐层向高时间层传播的累积,得到新的数值解总误差上界的理论近似公式,以及最优格距和最优时间步长的理论公式.通过数值算例验证了所得公式的可靠性。然后,发现了两种不同机器精度下最优时间步长之比满足的一个仅与机器精度有关的普适关系.最后,理论验证了在网格比固定的情况下,此问题满足数值计算的不确定性原理,以及在机器有限精度下最优时间步长的必然存在。

CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME FOR ISENTROPIC GAS DYNAMICS (Ⅰ)

A compactness frame of the Lax-Friedrichs scheme for the equations of gas dynamics is obtained by using some embedding theorems and an analysis of the difference scheme and the weak entropy.

On Accuracy of Lax-Friedrichs Scheme for Discontinuous Solutions of Convection Equations

Derive L-2-error bounds for Lax-Friedrichs schemes for discontinuous solutions oflinear hyperbolic convection equations.It is known that the Lax-Friedrichs scheme is a firstorder scheme.Analyzes convergent rate of the scheme through its modified equations andshows that the first order Lax-Friedrichs scheme to approach BV solutions of the convectionequation has L￣2-error bounds of O(△x￣(1/4)),where △x is the discrete mesh length.Nemericalexperiments are presented and numerical results justify the theoretical analysis.

A SUPPLEMENT TO THE PAPERS CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME FOR ISENTROPIC GAS DYNAMICS (Ⅱ)-(Ⅲ)

In the papers (Ⅱ)—(Ⅲ), we studied the global existence of the solutions for the isentropic equation of gas dynamics

CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX

The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is discontinuous. It is shown that a convergentsubsequence of approximations produced by the Lax-Friedrichs scheme converges tosuch an entropy solution, implying that the entire computed sequence converges.

Construction of a Computational Scheme for the Fuzzy HIV/AIDS Epidemic Model with a Nonlinear Saturated Incidence Rate

This work aimed to construct an epidemic model with fuzzy parameters.Since the classical epidemic model doesnot elaborate on the successful interaction of susceptible and infective people,the constructed fuzzy epidemicmodel discusses the more detailed versions of the interactions between infective and susceptible people.Thenext-generation matrix approach is employed to find the reproduction number of a deterministic model.Thesensitivity analysis and local stability analysis of the systemare also provided.For solving the fuzzy epidemic model,a numerical scheme is constructed which consists of three time levels.The numerical scheme has an advantage overthe existing forward Euler scheme for determining the conditions of getting the positive solution.The establishedscheme also has an advantage over existing non-standard finite difference methods in terms of order of accuracy.The stability of the scheme for the considered fuzzy model is also provided.From the plotted results,it can beobserved that susceptible people decay by rising interaction parameters.

On the Use of Monotonicity-Preserving Interpolatory Techniques in Multilevel Schemes for Balance Laws

Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock capturing numerical schemes.Because they do not involve any special data structure,and do not induce savings in memory requirements,they are easily implemented on existing codes and are recommended for 1D and 2D simulations when intensive testing is required.The multilevel technique can also be applied to balance laws,but in this case,numerical errors may be induced by the technique.We present a series of numerical tests that point out that the use of monotonicity-preserving interpolatory techniques eliminates the numerical errors observed when using the usual 4-point centered Lagrange interpolation,and leads to a more robust multilevel code for balance laws,while maintaining the efficiency rates observed forhyperbolic conservation laws.

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case.These kinetic models depend on a small parameter that can be seen as a"Knudsen"number.The method is asymptotic preserving in this Knudsen number.Also,the computational costs of the method are of the same order of a fully explicit scheme.This work is the extension of Abgrall et al.(2022)[3]to multidimensional systems.We have assessed our method on several problems for two-dimensional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

S-scheme heterojunction with ultrafast interfacial electron transfer for artificial photosynthesis

The rapid development of the global economy has led to the over-exploitation and burning of fossil fuels,causing a severe energy crisis and continuous CO_(2) emissions.Although solar energy is a clean and renewable resource,it faces significant diurnal and seasonal variations and is difficult to store[1-4].Converting solar energy into storable chemical energy through photocatalysis is an effective way to address both energy scarcity and environmental issues.Photocatalytic CO_(2) reduction,with the development of high-efficiency photocatalysts as the key,offers a clean and environmentally friendly method to convert CO_(2) into valuable hydrocarbon fuels,providing a viable solution to the global energy crisis and climate change[5,6].

Orbit Weighting Scheme in the Context of Vector Space Information Retrieval

This study introduces the Orbit Weighting Scheme(OWS),a novel approach aimed at enhancing the precision and efficiency of Vector Space information retrieval(IR)models,which have traditionally relied on weighting schemes like tf-idf and BM25.These conventional methods often struggle with accurately capturing document relevance,leading to inefficiencies in both retrieval performance and index size management.OWS proposes a dynamic weighting mechanism that evaluates the significance of terms based on their orbital position within the vector space,emphasizing term relationships and distribution patterns overlooked by existing models.Our research focuses on evaluating OWS’s impact on model accuracy using Information Retrieval metrics like Recall,Precision,InterpolatedAverage Precision(IAP),andMeanAverage Precision(MAP).Additionally,we assessOWS’s effectiveness in reducing the inverted index size,crucial for model efficiency.We compare OWS-based retrieval models against others using different schemes,including tf-idf variations and BM25Delta.Results reveal OWS’s superiority,achieving a 54%Recall and 81%MAP,and a notable 38%reduction in the inverted index size.This highlights OWS’s potential in optimizing retrieval processes and underscores the need for further research in this underrepresented area to fully leverage OWS’s capabilities in information retrieval methodologies.

Deployment Dynamic Modeling and Driving Schemes for a Ring-Truss Deployable Antenna

Mesh reflector antennas are widely used in space tasks owing to their light weight,high surface accuracy,and large folding ratio.They are stowed during launch and then fully deployed in orbit to form a mesh reflector that transmits signals.Smooth deployment is essential for duty services;therefore,accurate and efficient dynamic modeling and analysis of the deployment process are essential.One major challenge is depicting time-varying resistance of the cable network and capturing the cable-truss coupling behavior during the deployment process.This paper proposes a general dynamic analysis methodology for cable-truss coupling.Considering the topological diversity and geometric nonlinearity,the cable network's equilibrium equation is derived,and an explicit expression of the time-varying tension of the boundary cables,which provides the main resistance in truss deployment,is obtained.The deployment dynamic model is established,which considers the coupling effect between the soft cables and deployable truss.The effects of the antenna's driving modes and parameters on the dynamic deployment performance were investigated.A scaled prototype was manufactured,and the deployment experiment was conducted to verify the accuracy of the proposed modeling method.The proposed methodology is suitable for general cable antennas with arbitrary topologies and parameters,providing theoretical guidance for the dynamic performance evaluation of antenna driving schemes.

Improvement and security analysis of multi-ring discrete modulation continuous variable quantum secret sharing scheme

In order to avoid the complexity of Gaussian modulation and the problem that the traditional point-to-point communication DM-CVQKD protocol cannot meet the demand for multi-user key sharing at the same time, we propose a multi-ring discrete modulation continuous variable quantum key sharing scheme(MR-DM-CVQSS). In this paper, we primarily compare single-ring and multi-ring M-symbol amplitude and phase-shift keying modulations. We analyze their asymptotic key rates against collective attacks and consider the security key rates under finite-size effects. Leveraging the characteristics of discrete modulation, we improve the quantum secret sharing scheme. Non-dealer participants only require simple phase shifters to complete quantum secret sharing. We also provide the general design of the MR-DM-CVQSS protocol.We conduct a comprehensive analysis of the improved protocol's performance, confirming that the enhancement through multi-ring M-PSK allows for longer-distance quantum key distribution. Additionally, it reduces the deployment complexity of the system, thereby increasing the practical value.

A Non-Parametric Scheme for Identifying Data Characteristic Based on Curve Similarity Matching

For accurately identifying the distribution charac-teristic of Gaussian-like noises in unmanned aerial vehicle(UAV)state estimation,this paper proposes a non-parametric scheme based on curve similarity matching.In the framework of the pro-posed scheme,a Parzen window(kernel density estimation,KDE)method on sliding window technology is applied for roughly esti-mating the sample probability density,a precise data probability density function(PDF)model is constructed with the least square method on K-fold cross validation,and the testing result based on evaluation method is obtained based on some data characteristic analyses of curve shape,abruptness and symmetry.Some com-parison simulations with classical methods and UAV flight exper-iment shows that the proposed scheme has higher recognition accuracy than classical methods for some kinds of Gaussian-like data,which provides better reference for the design of Kalman filter(KF)in complex water environment.

A Neural-network-based Alternative Scheme to Include Nonhydrostatic Processes in an Atmospheric Dynamical Core

Here,a nonhydrostatic alternative scheme(NAS)is proposed for the grey zone where the nonhydrostatic impact on the atmosphere is evident but not large enough to justify the necessity to include an implicit nonhydrostatic solver in an atmospheric dynamical core.The NAS is designed to replace this solver,which can be incorporated into any hydrostatic models so that existing well-developed hydrostatic models can effectively serve for a longer time.Recent advances in machine learning(ML)provide a potential tool for capturing the main complicated nonlinear-nonhydrostatic relationship.In this study,an ML approach called a neural network(NN)was adopted to select leading input features and develop the NAS.The NNs were trained and evaluated with 12-day simulation results of dry baroclinic-wave tests by the Weather Research and Forecasting(WRF)model.The forward time difference of the nonhydrostatic tendency was used as the target variable,and the five selected features were the nonhydrostatic tendency at the last time step,and four hydrostatic variables at the current step including geopotential height,pressure in two different forms,and potential temperature,respectively.Finally,a practical NAS was developed with these features and trained layer by layer at a 20-km horizontal resolution,which can accurately reproduce the temporal variation and vertical distribution of the nonhydrostatic tendency.Corrected by the NN-based NAS,the improved hydrostatic solver at different horizontal resolutions can run stably for at least one month and effectively reduce most of the nonhydrostatic errors in terms of system bias,anomaly root-mean-square error,and the error of the wave spatial pattern,which proves the feasibility and superiority of this scheme.

Verifiable quantum secret sharing scheme based on orthogonal product states

In the domain of quantum cryptography,the implementation of quantum secret sharing stands as a pivotal element.In this paper,we propose a novel verifiable quantum secret sharing protocol using the d-dimensional product state and Lagrange interpolation techniques.This protocol is initiated by the dealer Alice,who initially prepares a quantum product state,selected from a predefined set of orthogonal product states within the C~d■C~d framework.Subsequently,the participants execute unitary operations on this product state to recover the underlying secret.Furthermore,we subject the protocol to a rigorous security analysis,considering both eavesdropping attacks and potential dishonesty from the participants.Finally,we conduct a comparative analysis of our protocol against existing schemes.Our scheme exhibits economies of scale by exclusively employing quantum product states,thereby realizing significant cost-efficiency advantages.In terms of access structure,we adopt a(t, n)-threshold architecture,a strategic choice that augments the protocol's practicality and suitability for diverse applications.Furthermore,our protocol includes a rigorous integrity verification mechanism to ensure the honesty and reliability of the participants throughout the execution of the protocol.

Quantum-Resistant Multi-Feature Attribute-Based Proxy Re-Encryption Scheme for Cloud Services

Cloud-based services have powerful storage functions and can provide accurate computation.However,the question of how to guarantee cloud-based services access control and achieve data sharing security has always been a research highlight.Although the attribute-based proxy re-encryption(ABPRE)schemes based on number theory can solve this problem,it is still difficult to resist quantum attacks and have limited expression capabilities.To address these issues,we present a novel linear secret sharing schemes(LSSS)matrix-based ABPRE scheme with the fine-grained policy on the lattice in the research.Additionally,to detect the activities of illegal proxies,homomorphic signature(HS)technology is introduced to realize the verifiability of re-encryption.Moreover,the non-interactivity,unidirectionality,proxy transparency,multi-use,and anti-quantum attack characteristics of our system are all advantageous.Besides,it can efficiently prevent the loss of processing power brought on by repetitive authorisation and can enable precise and safe data sharing in the cloud.Furthermore,under the standard model,the proposed learning with errors(LWE)-based scheme was proven to be IND-sCPA secure.

A Practical Regular LDPC Coded Scheme for Physical-Layer Information Security

In this paper,we aim to design a practical low complexity low-density parity-check(LDPC)coded scheme to build a secure open channel and protect information from eavesdropping.To this end,we first propose a punctured LDPC coded scheme,where the information bits in a codeword are punctured and only the parity check bits are transmitted to the receiver.We further propose a notion of check node type distribution and derive multi-edge type extrinsic information transfer functions to estimate the security performance,instead of the well-known weak metric bit error rate.We optimize the check node type distribution in terms of the signal-to-noise ratio(SNR)gap and modify the progressive edge growth algorithm to design finite-length codes.Numerical results show that our proposed scheme can achieve a lower computational complexity and a smaller security gap,compared to the existing scrambling and puncturing schemes.

A hybrid ventilation scheme applied to bidirectional excavation tunnel construction with a long inclined shaft

The breakage and bending of ducts result in a difficulty to cope with ventilation issues in bidirectional excavation tunnels with a long inclined shaft using a single ventilation method based on ducts.To discuss the hybrid ventilation system applied in bidirectional excavation tunnels with a long inclined shaft,this study has established a full-scale computational fluid dynamics model based on field tests,the Poly-Hexcore method,and the sliding mesh technique.The distribution of wind speed,temperature field,and CO in the tunnel are taken as indices to compare the ventilation efficiency of three ventilation systems(duct,duct-ventilation shaft,duct–ventilated shaft-axial fan).The results show that the hybrid ventilation scheme based on duct-ventilation shaft–axial fan performs the best among the three ventilation systems.Compared to the duct,the wind speed and cooling rate in the tunnel are enhanced by 7.5%–30.6%and 14.1%–17.7%,respectively,for the duct-vent shaft-axial fan condition,and the volume fractions of CO are reduced by 26.9%–73.9%.This contributes to the effective design of combined ventilation for bidirectional excavation tunnels with an inclined shaft,ultimately improving the air quality within the tunnel.

Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.