This paper presents an extensive survey of the most commonly used tools for diagnosing unbalanced flow in the atmosphere, namely the Lagrangian Rossby number, Psi vector, divergence equation, nonlinear balance equatio...This paper presents an extensive survey of the most commonly used tools for diagnosing unbalanced flow in the atmosphere, namely the Lagrangian Rossby number, Psi vector, divergence equation, nonlinear balance equation, generalized omega-equation, and departure from fields obtained by potential vorticity (PV) inversion. The basic thoery, assumptions as well as implementation and limitations for each of the tools are all discussed. These tools are applied to high—resolution mesoscale model data to assess the role of unbalanced dynamics in the generation of a mesoscale gravity wave event over the East Coast of the United States. Comparison of these tools in this case study shows that these various methods agree to a large extent with each other though they differ in details. Key words Unbalanced flow - Geostrophic adjustment - Gravity waves - Nonlinear balance equation - Potential vorticity inversion - Omega equations - Rossby number This research was conducted under support from NSF grant ATM-9700626 of the United States. The numerical computations described herein were performed on the Cray T90 at the North Carolina Supercomputing Center and the Cray supercomputer at the NCAR Scientific Computing Division, which also provided the initialization fields for the MM5. Thanks are extended to Mark Stoelinga at University of Washington for the RIP post-processing package.展开更多
A single-server queueing system with preemptive access is considered.Each customer has one attempt to enter the system at its working interval[0,T].As soon as the customer request enters the system,the server immediat...A single-server queueing system with preemptive access is considered.Each customer has one attempt to enter the system at its working interval[0,T].As soon as the customer request enters the system,the server immediately starts the service.But when the next request arrives in the system,the previous one leaves the system even he has not finished his service yet.We study a non-cooperative game in which the customers wish to maximize their probability of obtaining service within a certain period of time.We characterize the Nash equilibrium and the price of anarchy,which is defined as the ratio between the optimal and equilibrium social utility.Two models are considered.In the first model the number of players is fixed,while in the second it is random and obeys the Poisson distribution.We demonstrate that there exists a unique symmetric equilibrium for both models.Finally,we calculate the price of anarchy for both models and show that the price of anarchy is not monotone with respect to the number of customers.展开更多
This paper verifies the low Mach number limit of the non-isentropic compressible magnetohydrodynamic(MHD)equations with or without the magnetic diffusion in a three-dimensional bounded domain when the temperature vari...This paper verifies the low Mach number limit of the non-isentropic compressible magnetohydrodynamic(MHD)equations with or without the magnetic diffusion in a three-dimensional bounded domain when the temperature variation is large but finite.The uniform estimates of strong solutions are established in a short time interval independent of the Mach number,provided that the slip boundary condition for the velocity and the Neumann boundary condition for the temperature are imposed and the initial data is well-prepared.展开更多
For any fixed Alfvén number, the local well-posedness is proved for the equations of threedimensional ideal incompressible magneto-hydrodynamics in a domain with boundaries. Under appropriate conditions, a smooth...For any fixed Alfvén number, the local well-posedness is proved for the equations of threedimensional ideal incompressible magneto-hydrodynamics in a domain with boundaries. Under appropriate conditions, a smooth solution is shown to exist in a time interval independent of the Alfvén number, and the solutions of the original system tend to the solutions of a two-dimensional Euler flow coupled with a linear transport equation as the Alfvén number goes to zero.展开更多
The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing ...The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.展开更多
The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by...The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by Pierre de Fermat in the seventeenth century. It is widely regarded that no elementary proof of this theorem exists. The author provides evidence to dispel this belief.展开更多
The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,wh...The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.展开更多
The superconvergence of Multhopp′s discretization for the solution to the normal wash integral equation for the flow past a curved plate was theoretically analyzed and numerically examined. The Multhopp′s discretiza...The superconvergence of Multhopp′s discretization for the solution to the normal wash integral equation for the flow past a curved plate was theoretically analyzed and numerically examined. The Multhopp′s discretization also has a superconvergence behavior in simulating the vortex sheet evolution. An improved Multhopp′s method was suggested and applied to the numerical simulation of a periodical evolution of a flat vortex sheet. To validate the results obtained by the inviscid vortex method, the initial-value problem for the development of a shear layer at large Reynolds number was numerically investigated by solving the two-dimensional incompressible Navier-Stokes equations.展开更多
The literature model studied in this article describes bubble formation and growth in a highly viscous polymer liquid with support of a gaseous matter dispersed under pressure before foaming. The foam growth is induce...The literature model studied in this article describes bubble formation and growth in a highly viscous polymer liquid with support of a gaseous matter dispersed under pressure before foaming. The foam growth is induced by the application of vacuum and mass transport of volatile components dissolved in the polymer liquid. Based on this literature model, aeration processes are calculated for intermediate viscosity and low viscosity biological systems, as they are of interest for biomatter foams, in particular for food foams in industrial processes. At the end of this article, the numerical results are presented and discussed.展开更多
The raging COVID-19 pandemic is arguably the most important threat to global health presently.Although there is currently a vaccine,preventive measures have been proposed to reduce the spread of infection but the effi...The raging COVID-19 pandemic is arguably the most important threat to global health presently.Although there is currently a vaccine,preventive measures have been proposed to reduce the spread of infection but the efficacy of these interventions,and their likely impact on the number of COVID-19 infections is unknown.In this study,we proposed the SEIQHRS model(susceptible-exposed-infectious-quarantine-hospitalized-recovered-susceptible)model that predicts the trajectory of the epidemic to help plan an effective control strategy for COVID-19 in Ghana.We provided a short-term forecast of the early phase of the epidemic trajectory in Ghana using the generalized growth model.We estimated the effective basic Reproductive number Re in real-time using three different estimation procedures and simulated worse case epidemic scenarios and the impact of integrated individual and government interventions on the epidemic in the long term using compartmental models.The maximum likelihood estimates of Re and the corresponding 95%confidence interval was 2.04[95%CI:1.82e2.27;12th March-7th April 2020].The Re estimate using the exponential growth method was 2.11[95%CI:2.00e2.24]within the same period.The Re estimate using time-dependent(TD)method showed a gradual decline of the Effective Reproductive Number since March 12,2020 when the first 2 index cases were recorded but the rate of transmission remains high(TD:Re=2.52;95%CI:[1.87e3.49]).The current estimate of Re based on the TD method is 1.74[95%CI:1.41 e2.10;(13th May 2020)]but with comprehensive integrated government and individual level interventions,the Re could reduce to 0.5 which is an indication of the epidemic dying out in the general population.Our results showed that enhanced government and individual-level interventions and the intensity of media coverage could have a substantial effect on suppressing transmission of new COVID-19 cases and reduced death rates in Ghana until such a time that a potent vaccine or drug is discovered.展开更多
We present new large time step methods for the shallow water flows in the lowFroude number limit.In order to take into accountmultiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split ...We present new large time step methods for the shallow water flows in the lowFroude number limit.In order to take into accountmultiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection.We propose to approximate fast linear waves implicitly in time and in space bymeans of a genuinely multidimensional evolution operator.On the other hand,we approximate nonlinear advection part explicitly in time and in space bymeans of themethod of characteristics or some standard numerical flux function.Time integration is realized by the implicit-explicit(IMEX)method.We apply the IMEX Euler scheme,two step Runge Kutta Cranck Nicolson scheme,as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit.Numerical experiments demonstrate stability,accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.展开更多
An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density...An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficultyin the low Mach regime. The key idea of our all speed scheme is the special semiimplicit time discretization, in which the low Mach number stiff term is divided intotwo parts, one being treated explicitly and the other one implicitly. Moreover, the fluxof the density equation is also treated implicitly and an elliptic type equation is derivedto obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs tobe solved implicitly which reduces much computational cost. We develop this semiimplicit time discretization in the framework of a first order Local Lax-Friedrichs (orRusanov) scheme and numerical tests are displayed to demonstrate its performances.展开更多
This work obtains the disguise version of exact solitary wave solutions of the generalized(2+1)-dimensk>nal Zakharov-Kuznetsov-Benjamin-Bona-Mahony and the regularized long wave equation with some free parame...This work obtains the disguise version of exact solitary wave solutions of the generalized(2+1)-dimensk>nal Zakharov-Kuznetsov-Benjamin-Bona-Mahony and the regularized long wave equation with some free parameters via modified simple equation method(MSE).Usually the method does not give any solution if the balance number is more than one,but we apply MSE method successfully in different way to carry out the solutions of nonlinear evolution equation with balance number two.Finally some graphical results of the velocity profiles are presented for different values of the material constants.It is shown that this method,without help of any symbolic computation,provide a straightforward and powerful mathematical tool for solving nonlinear evolution equation.展开更多
文摘This paper presents an extensive survey of the most commonly used tools for diagnosing unbalanced flow in the atmosphere, namely the Lagrangian Rossby number, Psi vector, divergence equation, nonlinear balance equation, generalized omega-equation, and departure from fields obtained by potential vorticity (PV) inversion. The basic thoery, assumptions as well as implementation and limitations for each of the tools are all discussed. These tools are applied to high—resolution mesoscale model data to assess the role of unbalanced dynamics in the generation of a mesoscale gravity wave event over the East Coast of the United States. Comparison of these tools in this case study shows that these various methods agree to a large extent with each other though they differ in details. Key words Unbalanced flow - Geostrophic adjustment - Gravity waves - Nonlinear balance equation - Potential vorticity inversion - Omega equations - Rossby number This research was conducted under support from NSF grant ATM-9700626 of the United States. The numerical computations described herein were performed on the Cray T90 at the North Carolina Supercomputing Center and the Cray supercomputer at the NCAR Scientific Computing Division, which also provided the initialization fields for the MM5. Thanks are extended to Mark Stoelinga at University of Washington for the RIP post-processing package.
基金supported by the Russian Science Foundation(No.22-11-20015,https://rscf.ru/project/22-11-20015/)jointly with support of the authorities of the Republic of Karelia with funding from the Venture Investment Foundation of the Republic of Karelia.Also the research was supported by the National Natural Science Foundation of China(No.72171126).
文摘A single-server queueing system with preemptive access is considered.Each customer has one attempt to enter the system at its working interval[0,T].As soon as the customer request enters the system,the server immediately starts the service.But when the next request arrives in the system,the previous one leaves the system even he has not finished his service yet.We study a non-cooperative game in which the customers wish to maximize their probability of obtaining service within a certain period of time.We characterize the Nash equilibrium and the price of anarchy,which is defined as the ratio between the optimal and equilibrium social utility.Two models are considered.In the first model the number of players is fixed,while in the second it is random and obeys the Poisson distribution.We demonstrate that there exists a unique symmetric equilibrium for both models.Finally,we calculate the price of anarchy for both models and show that the price of anarchy is not monotone with respect to the number of customers.
基金supported by National Natural Science Foundation of China(Grant Nos.11971477,12131007 and 11761141008)the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(Grant No.18XNLG30)。
文摘This paper verifies the low Mach number limit of the non-isentropic compressible magnetohydrodynamic(MHD)equations with or without the magnetic diffusion in a three-dimensional bounded domain when the temperature variation is large but finite.The uniform estimates of strong solutions are established in a short time interval independent of the Mach number,provided that the slip boundary condition for the velocity and the Neumann boundary condition for the temperature are imposed and the initial data is well-prepared.
基金supported by the Israel Science Foundation-National Natural Science Foundation of China Joint Research Program (Grant No. 11761141008)supported by the National Basic Research Program of China (Grant No. 2014CB745002)+2 种基金National Natural Science Foundation of China (Grant No. 11631008)supported by National Natural Science Foundation of China (Grant Nos. 11571046, 11471028 and 11671225)Beijing Natural Science Foundation (Grant No. 1182004)
文摘For any fixed Alfvén number, the local well-posedness is proved for the equations of threedimensional ideal incompressible magneto-hydrodynamics in a domain with boundaries. Under appropriate conditions, a smooth solution is shown to exist in a time interval independent of the Alfvén number, and the solutions of the original system tend to the solutions of a two-dimensional Euler flow coupled with a linear transport equation as the Alfvén number goes to zero.
文摘The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.
文摘The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by Pierre de Fermat in the seventeenth century. It is widely regarded that no elementary proof of this theorem exists. The author provides evidence to dispel this belief.
基金the National Numerical Wind Tunnel Project(No.NNW2018-ZT4A08)the National Natural Science Foundation of China(Nos.11872323 and 11971025)the Natural Science Foundation of Fujian Province(No.2019J06002)。
文摘The computation of compressible flows at all Mach numbers is a very challenging problem.An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime,while it can deal with stiffness and accuracy in the low Mach number regime.This paper designs a high order semi-implicit weighted compact nonlinear scheme(WCNS)for the all-Mach isentropic Euler system of compressible gas dynamics.To avoid severe Courant-Friedrichs-Levy(CFL)restrictions for low Mach flows,the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components.A third-order implicit-explicit(IMEX)method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives.The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit.One-and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.
基金Project supported by the National Natural Science Foundation of China.(No.1 93 72 0 60 )
文摘The superconvergence of Multhopp′s discretization for the solution to the normal wash integral equation for the flow past a curved plate was theoretically analyzed and numerically examined. The Multhopp′s discretization also has a superconvergence behavior in simulating the vortex sheet evolution. An improved Multhopp′s method was suggested and applied to the numerical simulation of a periodical evolution of a flat vortex sheet. To validate the results obtained by the inviscid vortex method, the initial-value problem for the development of a shear layer at large Reynolds number was numerically investigated by solving the two-dimensional incompressible Navier-Stokes equations.
文摘The literature model studied in this article describes bubble formation and growth in a highly viscous polymer liquid with support of a gaseous matter dispersed under pressure before foaming. The foam growth is induced by the application of vacuum and mass transport of volatile components dissolved in the polymer liquid. Based on this literature model, aeration processes are calculated for intermediate viscosity and low viscosity biological systems, as they are of interest for biomatter foams, in particular for food foams in industrial processes. At the end of this article, the numerical results are presented and discussed.
文摘The raging COVID-19 pandemic is arguably the most important threat to global health presently.Although there is currently a vaccine,preventive measures have been proposed to reduce the spread of infection but the efficacy of these interventions,and their likely impact on the number of COVID-19 infections is unknown.In this study,we proposed the SEIQHRS model(susceptible-exposed-infectious-quarantine-hospitalized-recovered-susceptible)model that predicts the trajectory of the epidemic to help plan an effective control strategy for COVID-19 in Ghana.We provided a short-term forecast of the early phase of the epidemic trajectory in Ghana using the generalized growth model.We estimated the effective basic Reproductive number Re in real-time using three different estimation procedures and simulated worse case epidemic scenarios and the impact of integrated individual and government interventions on the epidemic in the long term using compartmental models.The maximum likelihood estimates of Re and the corresponding 95%confidence interval was 2.04[95%CI:1.82e2.27;12th March-7th April 2020].The Re estimate using the exponential growth method was 2.11[95%CI:2.00e2.24]within the same period.The Re estimate using time-dependent(TD)method showed a gradual decline of the Effective Reproductive Number since March 12,2020 when the first 2 index cases were recorded but the rate of transmission remains high(TD:Re=2.52;95%CI:[1.87e3.49]).The current estimate of Re based on the TD method is 1.74[95%CI:1.41 e2.10;(13th May 2020)]but with comprehensive integrated government and individual level interventions,the Re could reduce to 0.5 which is an indication of the epidemic dying out in the general population.Our results showed that enhanced government and individual-level interventions and the intensity of media coverage could have a substantial effect on suppressing transmission of new COVID-19 cases and reduced death rates in Ghana until such a time that a potent vaccine or drug is discovered.
基金supported by the German Science Foundation under the grants LU 1470/2-2 and No 361/3-2.The second author has been supported by the Alexander-von-Humboldt Foundation through a postdoctoral fellowship.M.L.and G.B.would like to thank Dr.Leonid Yelash(JGU Mainz)for fruitful discussions.
文摘We present new large time step methods for the shallow water flows in the lowFroude number limit.In order to take into accountmultiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection.We propose to approximate fast linear waves implicitly in time and in space bymeans of a genuinely multidimensional evolution operator.On the other hand,we approximate nonlinear advection part explicitly in time and in space bymeans of themethod of characteristics or some standard numerical flux function.Time integration is realized by the implicit-explicit(IMEX)method.We apply the IMEX Euler scheme,two step Runge Kutta Cranck Nicolson scheme,as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit.Numerical experiments demonstrate stability,accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.
基金supported by the French"Commissariat´a l’Energie Atomique(CEA)"(Centre de Saclay)in the frame of the contract"ASTRE",#SAV 34160the Marie Curie Actions of the European Commission in the frame of the DEASE project(MESTCT-2005-021122)。
文摘An all speed scheme for the Isentropic Euler equations is presented in thispaper. When the Mach number tends to zero, the compressible Euler equations converge to their incompressible counterpart, in which the density becomes a constant. Increasing approximation errors and severe stability constraints are the main difficultyin the low Mach regime. The key idea of our all speed scheme is the special semiimplicit time discretization, in which the low Mach number stiff term is divided intotwo parts, one being treated explicitly and the other one implicitly. Moreover, the fluxof the density equation is also treated implicitly and an elliptic type equation is derivedto obtain the density. In this way, the correct limit can be captured without requesting the mesh size and time step to be smaller than the Mach number. Compared withprevious semi-implicit methods [11,13,29], firstly, nonphysical oscillations can be suppressed by choosing proper parameter, besides, only a linear elliptic equation needs tobe solved implicitly which reduces much computational cost. We develop this semiimplicit time discretization in the framework of a first order Local Lax-Friedrichs (orRusanov) scheme and numerical tests are displayed to demonstrate its performances.
文摘This work obtains the disguise version of exact solitary wave solutions of the generalized(2+1)-dimensk>nal Zakharov-Kuznetsov-Benjamin-Bona-Mahony and the regularized long wave equation with some free parameters via modified simple equation method(MSE).Usually the method does not give any solution if the balance number is more than one,but we apply MSE method successfully in different way to carry out the solutions of nonlinear evolution equation with balance number two.Finally some graphical results of the velocity profiles are presented for different values of the material constants.It is shown that this method,without help of any symbolic computation,provide a straightforward and powerful mathematical tool for solving nonlinear evolution equation.
