This study examines the stability regimes of three-dimensional interfacial gravity waves.The numerical results of the linear stability analysis extend the three-dimensional surface waves results of Ioualalen and Khari...This study examines the stability regimes of three-dimensional interfacial gravity waves.The numerical results of the linear stability analysis extend the three-dimensional surface waves results of Ioualalen and Kharif(1994)to three-dimensional interfacial waves.An approach of the collocation type has been developed for this purpose.The equations of motion are reduced to an eigenvalue problem where the perturbations are spectrally decomposed into normal modes.The results obtained showed that the density ratio plays a stabilizing factor.In addition,the dominant instability is of three-dimensional structure,and it belongs to class I for all values of density ratio.展开更多
This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximat...This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximation effects than those of prevailing methods.In principle,the IBM can be applied for bounding more bounded smooth functions and their integrals as well,and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.展开更多
Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in m...Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padéapproximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that(1) they approximate the entire solutions quite well from the core to infinity;(2) higher-order Padé approximate solutions have higher accuracy;(3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with 87Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.展开更多
This paper proposes a new approach that combines the reduced differential transform method (RDTM), a resummation method based on the Yang transform, and a Padé approximant to the kinetically reduced local Navier-...This paper proposes a new approach that combines the reduced differential transform method (RDTM), a resummation method based on the Yang transform, and a Padé approximant to the kinetically reduced local Navier-Stokes equation to find approximate solutions to the problem of lid-driven square cavity flow. The new approach, called PYRDM, considerably improves the convergence rate of the truncated series solution of RDTM and also is based on a simple process that yields highly precise estimates. The numerical results achieved by this method are compared to earlier studies’ results. Our results indicate that this method is more efficient and precise in generating analytic solutions. Furthermore, it provides highly precise solutions with good convergence that is simple to apply for great Reynolds and low Mach numbers. Moreover, the new solution’ graphs demonstrate the new approach’s validity, usefulness, and necessity.展开更多
In last decades, several algorithms were developed for fast evaluation of some elementary functions with very large arguments, for example for multiplication of million-digit integers. The present paper introduces a n...In last decades, several algorithms were developed for fast evaluation of some elementary functions with very large arguments, for example for multiplication of million-digit integers. The present paper introduces a new fast iterative method for computing values ?with high accuracy, for fixed ?and . The method is based on compound means and Padé approximations.展开更多
In this paper,we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant.The virial expansion is ...In this paper,we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant.The virial expansion is a high-temperature and low-density expansion and in practice,often,only the first several virial coefficients can be obtained.For Bose gases,we determine the BEC phase transition from a truncated virial expansion.For Fermi gases,we recover the low-temperature and high-density result from the virial expansion.展开更多
The dynamics of vortices in Bose-Einstein condensates of dilute cold atoms can be well formulated by Gross-Pitaevskii equation.To better understand the properties of vortices,a systematic method to solve the nonlinear...The dynamics of vortices in Bose-Einstein condensates of dilute cold atoms can be well formulated by Gross-Pitaevskii equation.To better understand the properties of vortices,a systematic method to solve the nonlinear differential equation for the vortex to very high precision is proposed.Through two-point Padéapproximants,these solutions are presented in terms of simple rational functions,which can be used in the simulation of vortex dynamics.The precision of the solutions is sensitive to the connecting parameter and the truncation orders.It can be improved significantly with a reasonable extension in the order of rational functions.The errors of the solutions and the limitation of two-point Padéapproximants are discussed.This investigation may shed light on the exact solution to the nonlinear vortex equation.展开更多
The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and cal...The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and calculation accuracy or order,it is still lack of systematic analysis conclusions.In this paper,according to the principle of differential quadrature method,it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature.Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method,it has been proved that the differential quadrature method is A-stable and s-stage s-order method.On this basis,in order to further improve the accuracy of the time domain differential quadrature method,a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Pad´e approximations.The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.展开更多
This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neuman...This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neumann-type ABCs(global in time)and high-order Pad´e approximate ABCs(local in time).These ABCs reformulate the original problem into an initial-boundary-value(IBV)problem on a bounded domain.For the global ABCs,we adopt a fast evolution to enhance computational efficiency and reduce memory storage.High order fully discrete schemes,both second-order in time and space,are given to discretize two reduced problems.Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.展开更多
文摘This study examines the stability regimes of three-dimensional interfacial gravity waves.The numerical results of the linear stability analysis extend the three-dimensional surface waves results of Ioualalen and Kharif(1994)to three-dimensional interfacial waves.An approach of the collocation type has been developed for this purpose.The equations of motion are reduced to an eigenvalue problem where the perturbations are spectrally decomposed into normal modes.The results obtained showed that the density ratio plays a stabilizing factor.In addition,the dominant instability is of three-dimensional structure,and it belongs to class I for all values of density ratio.
基金Supported by the National Natural Science Foundation of China(61672009,61502130).
文摘This paper presents an interpolation-based method(IBM)for approximating some trigonometric functions or their integrals as well.It provides two-sided bounds for each function,which also achieves much better approximation effects than those of prevailing methods.In principle,the IBM can be applied for bounding more bounded smooth functions and their integrals as well,and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.
基金Undergraduate Innovation and Entrepreneurship Program Grant No.S201910579797National Natural Science Foundation of China with Grant No.12005088,11847001,11747017+1 种基金Guangdong Basic and Applied Basic Research Foundation with Grant No.2021A1515010246supported by the Lingnan Normal University Project with Grant No.YL20200203,ZL1930。
文摘Quantized vortices are important topological excitations in Bose–Einstein condensates. The Gross–Pitaevskii equation is a widely accepted theoretical tool. High accuracy quantized-vortex solutions are desirable in many numerical and analytical studies. We successfully derive the Padéapproximate solutions for quantized vortices with winding numbers ω = 1, 2, 3, 4, 5, 6 in the context of the Gross–Pitaevskii equation for a uniform condensate. Compared with the numerical solutions, we find that(1) they approximate the entire solutions quite well from the core to infinity;(2) higher-order Padé approximate solutions have higher accuracy;(3) Padé approximate solutions for larger winding numbers have lower accuracy. The healing lengths of the quantized vortices are calculated and found to increase almost linearly with the winding number. Based on experiments performed with 87Rb cold atoms, the healing lengths of quantized vortices and the number of particles within the healing lengths are calculated, and they may be checked by experiment. Our results show that the Gross–Pitaevskii equation is capable of describing the structure of quantized vortices and physics at length scales smaller than the healing length.
文摘This paper proposes a new approach that combines the reduced differential transform method (RDTM), a resummation method based on the Yang transform, and a Padé approximant to the kinetically reduced local Navier-Stokes equation to find approximate solutions to the problem of lid-driven square cavity flow. The new approach, called PYRDM, considerably improves the convergence rate of the truncated series solution of RDTM and also is based on a simple process that yields highly precise estimates. The numerical results achieved by this method are compared to earlier studies’ results. Our results indicate that this method is more efficient and precise in generating analytic solutions. Furthermore, it provides highly precise solutions with good convergence that is simple to apply for great Reynolds and low Mach numbers. Moreover, the new solution’ graphs demonstrate the new approach’s validity, usefulness, and necessity.
基金supported by grant P201/12/2351 of the Czech Science Foundationby grant 01798/2011/RRC of the Moravian-Silesian regionby grant SGS08/PrF/2014 of the University of Ostrava.
文摘In last decades, several algorithms were developed for fast evaluation of some elementary functions with very large arguments, for example for multiplication of million-digit integers. The present paper introduces a new fast iterative method for computing values ?with high accuracy, for fixed ?and . The method is based on compound means and Padé approximations.
基金supported in part by The Fundamental Research Funds for the Central Universities under Grant No.2020JKF306Special Funds for theoretical physics Research Program of the NSFC under Grant No.11947124,and NSFC under Grant Nos.11575125 and 11675119。
文摘In this paper,we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant.The virial expansion is a high-temperature and low-density expansion and in practice,often,only the first several virial coefficients can be obtained.For Bose gases,we determine the BEC phase transition from a truncated virial expansion.For Fermi gases,we recover the low-temperature and high-density result from the virial expansion.
文摘The dynamics of vortices in Bose-Einstein condensates of dilute cold atoms can be well formulated by Gross-Pitaevskii equation.To better understand the properties of vortices,a systematic method to solve the nonlinear differential equation for the vortex to very high precision is proposed.Through two-point Padéapproximants,these solutions are presented in terms of simple rational functions,which can be used in the simulation of vortex dynamics.The precision of the solutions is sensitive to the connecting parameter and the truncation orders.It can be improved significantly with a reasonable extension in the order of rational functions.The errors of the solutions and the limitation of two-point Padéapproximants are discussed.This investigation may shed light on the exact solution to the nonlinear vortex equation.
基金support from the National Natural Science Foundation of China(NSFC)through its grant 51377098.
文摘The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and calculation accuracy or order,it is still lack of systematic analysis conclusions.In this paper,according to the principle of differential quadrature method,it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature.Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method,it has been proved that the differential quadrature method is A-stable and s-stage s-order method.On this basis,in order to further improve the accuracy of the time domain differential quadrature method,a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Pad´e approximations.The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.
基金This research is supported in part by the U.S.NSF grants DMS-1318586 and DMS-1315259AFOSR MURI Center for Material Failure Prediction Through Peridynamics,OSD/ARO/MURI W911NF-15-1-0562 on Fractional PDEs for Conservation Laws and Beyond:Theory,Numerics and Applicationsthe NSFC under grants 91430216 and the NSFC program for Scientific Research Center under program No.:U1530401.
文摘This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain.Two classes of artificial boundary conditions(ABCs)are designed,namely,nonlocal analog Dirichlet-to-Neumann-type ABCs(global in time)and high-order Pad´e approximate ABCs(local in time).These ABCs reformulate the original problem into an initial-boundary-value(IBV)problem on a bounded domain.For the global ABCs,we adopt a fast evolution to enhance computational efficiency and reduce memory storage.High order fully discrete schemes,both second-order in time and space,are given to discretize two reduced problems.Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.