Energy-conserving Hermite methods for solving Maxwell’s equations in dielectric and dispersive media are described and analyzed. In three space dimensions, methods of order 2m to 2m+2 require (m+1)^(3) degrees-of-fre...Energy-conserving Hermite methods for solving Maxwell’s equations in dielectric and dispersive media are described and analyzed. In three space dimensions, methods of order 2m to 2m+2 require (m+1)^(3) degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of m. We prove the stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special semi-norm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of the electromagnetic wave propagation over thousands of wavelengths.展开更多
A distinguished category of operational fluids,known as hybrid nanofluids,occupies a prominent role among various fluid types owing to its superior heat transfer properties.By employing a dovetail fin profile,this wor...A distinguished category of operational fluids,known as hybrid nanofluids,occupies a prominent role among various fluid types owing to its superior heat transfer properties.By employing a dovetail fin profile,this work investigates the thermal reaction of a dynamic fin system to a hybrid nanofluid with shape-based properties,flowing uniformly at a velocity U.The analysis focuses on four distinct types of nanoparticles,i.e.,Al2O3,Ag,carbon nanotube(CNT),and graphene.Specifically,two of these particles exhibit a spherical shape,one possesses a cylindrical form,and the final type adopts a platelet morphology.The investigation delves into the pairing of these nanoparticles.The examination employs a combined approach to assess the constructional and thermal exchange characteristics of the hybrid nanofluid.The fin design,under the specified circumstances,gives rise to the derivation of a differential equation.The given equation is then transformed into a dimensionless form.Notably,the Hermite wavelet method is introduced for the first time to address the challenge posed by a moving fin submerged in a hybrid nanofluid with shape-dependent features.To validate the credibility of this research,the results obtained in this study are systematically compared with the numerical simulations.The examination discloses that the highest heat flux is achieved when combining nanoparticles with spherical and platelet shapes.展开更多
We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids.Near boundaries we use t...We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids.Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approaches a structured Cartesian Hermite method.Unlike many other overset methods we do not need to add artificial dissipation but we find that the built-in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain the stability.By numerical experiments we demonstrate the stability,accuracy,efficiency,and the applicability of the methods to forward and inverse problems.展开更多
Based on the operator Hermite polynomials method（OHPM）, we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with s...Based on the operator Hermite polynomials method（OHPM）, we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.展开更多
We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Pois-son system written as a hyperbolic system using Hermite polynomials in the velocity vari-able.These schemes are designed to be syst...We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Pois-son system written as a hyperbolic system using Hermite polynomials in the velocity vari-able.These schemes are designed to be systematically as accurate as one wants with prov-able conservation of mass and possibly total energy.Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system.The proposed scheme employs the discontinuous Galerkin discretization for both the Vlasov and the Poisson equations,resulting in a consistent description of the distribu-tion function and the electric field.Numerical simulations are performed to verify the order of the accuracy and conservation properties.展开更多
Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Eul...Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.展开更多
By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials wh...By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.展开更多
We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m（x）,∞∑n=0sntm/n!m!H2n,2m（x,y）,and ∞∑n=0sntm/n!m!H2n＋l,2m＋k（x,y）.We employ the operator Herm...We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m（x）,∞∑n=0sntm/n!m!H2n,2m（x,y）,and ∞∑n=0sntm/n!m!H2n＋l,2m＋k（x,y）.We employ the operator Hermite polynomial method and the technique of integration within an ordered product of operators to solve these problems, which will be useful in constructing new optical field states.展开更多
The Hermite-Taylor method,introduced in 2005 by Goodrich et al.is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains.Unfortunately,its widespread use has been prevented by the ...The Hermite-Taylor method,introduced in 2005 by Goodrich et al.is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains.Unfortunately,its widespread use has been prevented by the lack of a systematic approach to implementing boundary conditions.In this paper we present the Hermite-Taylor correction function method(CFM),which provides exactly such a systematic approach for handling boundary conditions.Here we focus on Maxwell’s equations but note that the method is easily extended to other hyperbolic problems.展开更多
In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed...In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.展开更多
In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recen...In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function φ.They are now computed from the current value of φ and the previous spatial derivatives of φ. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear Hamiltonian, therefore any HamiltonJacobi equations with complex Hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameterε used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods.展开更多
By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-seri...By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min（m,n）∑l=0^min（m,n）n!m!（-1）^l/l!（n-1）!（m-l）!/Hn-l（x）y^m-l≡ n,m（x,y）.By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators（IWOP） we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.展开更多
In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stoch...In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stochastic Kudryashov–Sinelshchikov equation to deterministic partial differential equation. Also we have applied inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.展开更多
In this paper, we continue the discussion of [12] to establish the Hermite pseudospectral method with weight ω（x） = 1. As an application, we consider the pseudospectral approximation of the reaction-diffusion equat...In this paper, we continue the discussion of [12] to establish the Hermite pseudospectral method with weight ω（x） = 1. As an application, we consider the pseudospectral approximation of the reaction-diffusion equation on the whole line, we prove the existence of the approximate attractor and give the error estimate for the approximate solution.展开更多
The McKean-Vlasov filtering problem is a special kind of filtering problem,with the state and/or observation processes governed by McKean-Vlasov stochastic differential equations,which has extensive applications in va...The McKean-Vlasov filtering problem is a special kind of filtering problem,with the state and/or observation processes governed by McKean-Vlasov stochastic differential equations,which has extensive applications in various scenarios.In this paper,the authors will propose a novel numerical algorithm to solve the McKean-Vlasov filtering problem based on the Hermite spectral method under the framework of Yau-Yau algorithm.As the first approach to numerically solving the Duncan-Mortensen-Zakai equation associated with the McKean-Vlasov filtering problem,the proposed algorithm can provide accurate estimations of the conditional expectation and conditional probability density of the state process with a reasonable online computational complexity.The efficiency of the proposed algorithm is verified both theoretically and numerically in this paper.展开更多
基金funded in part by the National Science Foundation Grants DMS-2012296,DMS-2309687,and DMS-2210286.
文摘Energy-conserving Hermite methods for solving Maxwell’s equations in dielectric and dispersive media are described and analyzed. In three space dimensions, methods of order 2m to 2m+2 require (m+1)^(3) degrees-of-freedom per node for each field variable and can be explicitly marched in time with steps independent of m. We prove the stability for time steps limited only by domain-of-dependence requirements along with error estimates in a special semi-norm associated with the interpolation process. Numerical experiments are presented which demonstrate that Hermite methods of very high order enable the efficient simulation of the electromagnetic wave propagation over thousands of wavelengths.
文摘A distinguished category of operational fluids,known as hybrid nanofluids,occupies a prominent role among various fluid types owing to its superior heat transfer properties.By employing a dovetail fin profile,this work investigates the thermal reaction of a dynamic fin system to a hybrid nanofluid with shape-based properties,flowing uniformly at a velocity U.The analysis focuses on four distinct types of nanoparticles,i.e.,Al2O3,Ag,carbon nanotube(CNT),and graphene.Specifically,two of these particles exhibit a spherical shape,one possesses a cylindrical form,and the final type adopts a platelet morphology.The investigation delves into the pairing of these nanoparticles.The examination employs a combined approach to assess the constructional and thermal exchange characteristics of the hybrid nanofluid.The fin design,under the specified circumstances,gives rise to the derivation of a differential equation.The given equation is then transformed into a dimensionless form.Notably,the Hermite wavelet method is introduced for the first time to address the challenge posed by a moving fin submerged in a hybrid nanofluid with shape-dependent features.To validate the credibility of this research,the results obtained in this study are systematically compared with the numerical simulations.The examination discloses that the highest heat flux is achieved when combining nanoparticles with spherical and platelet shapes.
基金This work was supported in part by the National Science Foundation under Grant NSF-1913076.Any opinions,findings,and conclusions or recommendations expressed in this material are those of the author(s)and do not necessarily reflect the views of the National Science Foundation.
文摘We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids.Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approaches a structured Cartesian Hermite method.Unlike many other overset methods we do not need to add artificial dissipation but we find that the built-in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain the stability.By numerical experiments we demonstrate the stability,accuracy,efficiency,and the applicability of the methods to forward and inverse problems.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175113)
文摘Based on the operator Hermite polynomials method（OHPM）, we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.
基金the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under Grant Agreement No.633053.the Science Challenge Project(No.TZ2016002)+2 种基金the National Natural Science Foundation of China(No.11971025)the Natural Science Foundation of Fujian Province(No.2019J06002)the NSAF(No.U1630247)。
文摘We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Pois-son system written as a hyperbolic system using Hermite polynomials in the velocity vari-able.These schemes are designed to be systematically as accurate as one wants with prov-able conservation of mass and possibly total energy.Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system.The proposed scheme employs the discontinuous Galerkin discretization for both the Vlasov and the Poisson equations,resulting in a consistent description of the distribu-tion function and the electric field.Numerical simulations are performed to verify the order of the accuracy and conservation properties.
文摘Viscoelastic foundation plays a very important role in civil engineering. It can effectively disperse the structural load into the foundation soil and avoid the damage caused by the concentrated load. The model of Euler-Bernoulli beam on viscoelastic Pasternak foundation can be used to analyze the deformation and response of buildings under complex geological conditions. In this paper, we use Hermite finite element method to get the numerical approximation scheme for the vibration equation of viscoelastic Pasternak foundation beam. Convergence and error estimation are rigourously established. We prove that the fully discrete scheme has convergence order O(τ2+h4), where τis time step size and his space step size. Finally, we give four numerical examples to verify the validity of theoretical analysis.
基金supported by the National Natural Science Foundation of China(Grant No.11175113)the Fundamental Research Funds for the Central Universities of China(Grant No.WK2060140013)
文摘By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.
基金Project supported by the National Natural Science Foundation of China(Grnat No.11175113)the Fundamental Research Funds for the Central Universities of China(Grant No.WK2060140013)
文摘We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m（x）,∞∑n=0sntm/n!m!H2n,2m（x,y）,and ∞∑n=0sntm/n!m!H2n＋l,2m＋k（x,y）.We employ the operator Hermite polynomial method and the technique of integration within an ordered product of operators to solve these problems, which will be useful in constructing new optical field states.
基金supported in part by the Grant NSF-2208164 and 2210286.
文摘The Hermite-Taylor method,introduced in 2005 by Goodrich et al.is highly efficient and accurate when applied to linear hyperbolic systems on periodic domains.Unfortunately,its widespread use has been prevented by the lack of a systematic approach to implementing boundary conditions.In this paper we present the Hermite-Taylor correction function method(CFM),which provides exactly such a systematic approach for handling boundary conditions.Here we focus on Maxwell’s equations but note that the method is easily extended to other hyperbolic problems.
基金supported by the NSF (Grant No.DMS-1753581)supported by NSFC (Grant No.12071392).
文摘In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
基金This work was carried out when Y.Ren was visiting Department of Mathematics,The Ohio State University.The work of Y.Xing is partially supported by the NSF grant DMS-1753581The work of J.Qiu is partially supported by NSFC grant 12071392.
文摘In this paper, we combine the nonlinear HWENO reconstruction in [43] andthe fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the staticHamilton-Jacobi equations in a novel HWENO framework recently developed in [22].The proposed HWENO frameworks enjoys several advantages. First, compared withthe traditional HWENO framework, the proposed methods do not need to introduceadditional auxiliary equations to update the derivatives of the unknown function φ.They are now computed from the current value of φ and the previous spatial derivatives of φ. This approach saves the computational storage and CPU time, which greatlyimproves the computational efficiency of the traditional HWENO scheme. In addition,compared with the traditional WENO method, reconstruction stencil of the HWENOmethods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping methodis used to update the numerical approximation. It is an explicit method and doesnot involve the inverse operation of nonlinear Hamiltonian, therefore any HamiltonJacobi equations with complex Hamiltonian can be solved easily. It also resolves someknown issues, including that the iterative number is very sensitive to the parameterε used in the nonlinear weights, as observed in previous studies. Finally, to furtherreduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate thegood performance of the proposed fixed-point fast sweeping HWENO methods.
基金supported by the Natural Science Fund of Education Department of Anhui Province,China(Grant No.KJ2016A590)the Talent Foundation of Hefei University,China(Grant No.15RC11)the National Natural Science Foundation of China(Grant Nos.11247009 and 11574295)
文摘By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min（m,n）∑l=0^min（m,n）n!m!（-1）^l/l!（n-1）!（m-l）!/Hn-l（x）y^m-l≡ n,m（x,y）.By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators（IWOP） we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.
文摘In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stochastic Kudryashov–Sinelshchikov equation to deterministic partial differential equation. Also we have applied inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.
基金Supported by the National Natural Sciences Foundation of China (No. 10771142) and (No. 10671130)
文摘In this paper, we continue the discussion of [12] to establish the Hermite pseudospectral method with weight ω（x） = 1. As an application, we consider the pseudospectral approximation of the reaction-diffusion equation on the whole line, we prove the existence of the approximate attractor and give the error estimate for the approximate solution.
基金supported by the National Natural Science Foundation of China under Grant No.123B2020by Tsinghua University Education Foundation fund under Grant No.042202008.
文摘The McKean-Vlasov filtering problem is a special kind of filtering problem,with the state and/or observation processes governed by McKean-Vlasov stochastic differential equations,which has extensive applications in various scenarios.In this paper,the authors will propose a novel numerical algorithm to solve the McKean-Vlasov filtering problem based on the Hermite spectral method under the framework of Yau-Yau algorithm.As the first approach to numerically solving the Duncan-Mortensen-Zakai equation associated with the McKean-Vlasov filtering problem,the proposed algorithm can provide accurate estimations of the conditional expectation and conditional probability density of the state process with a reasonable online computational complexity.The efficiency of the proposed algorithm is verified both theoretically and numerically in this paper.
