In this article, we study the nonlinear stochastic heat equation in the spatial domain R^d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnega...In this article, we study the nonlinear stochastic heat equation in the spatial domain R^d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z^d to that on R^d.Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.展开更多
This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model ∂u/∂t=1/2△+u˙W on[0,∞)×R^(d) with d≥1 has a unique randomfield solutio...This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model ∂u/∂t=1/2△+u˙W on[0,∞)×R^(d) with d≥1 has a unique randomfield solution,where W(t,x)is a fractional Brownian sheet on[0,∞)×Rd and formally ˙W=∂d+1/∂t+∂x_(1)…∂x_(d)=W(t,x).When the noise W(t,x) is white in time,our condition is both necessary and sufficient when the initial data u(0,x)is bounded between two positive constants.When the noise is fractional in time with Hurst parameter H_(0)>1/2,our sufficient condition,which improves the known results in the literature,is different from the necessary one.展开更多
As one of the main governing equations in kinetic theory,the Boltzmann equation is widely utilized in aerospace,microscopic flow,etc.Its high-resolution simulation is crucial in these related areas.However,due to the ...As one of the main governing equations in kinetic theory,the Boltzmann equation is widely utilized in aerospace,microscopic flow,etc.Its high-resolution simulation is crucial in these related areas.However,due to the high dimensionality of the Boltzmann equation,high-resolution simulations are often difficult to achieve numerically.The moment method which was first proposed in Grad(Commun Pure Appl Math 2(4):331-407,1949)is among the popular numerical methods to achieve efficient high-resolution simulations.We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation,which effectively reduces the dimensionality of the problem.However,one of themain challenges is that it leads to an unclosed moment system,and closure is needed to obtain a closedmoment system.It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory.Other than the traditional human designs of closures,the machine learning-based approach has attracted much attention lately in Han et al.(Proc Natl Acad Sci USA 116(44):21983-21991,2019)and Huang et al.(J Non-Equilib Thermodyn 46(4):355-370,2021).In this work,we propose a machine learning-based method to derive a moment closure model for the Boltzmann-BGK equation.In particular,the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances,i.e.,the Galilean invariance,reflecting invariance,and scaling invariance,inherited from the original Boltzmann-BGK equation and playing an important role in the correct simulation of the Boltzmann equation.Numerical simulations on the 1D-1D examples including the smooth and discontinuous initial condition problems,Sod shock tube problem,the shock structure problems,and the 1D-3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.展开更多
We study efficient simulation of steady state for multi-dimensional rarefied gas flow,which is modeled by the Boltzmann equation with BGK-type collision term.A nonlinear multigrid solver is proposed to resolve the eff...We study efficient simulation of steady state for multi-dimensional rarefied gas flow,which is modeled by the Boltzmann equation with BGK-type collision term.A nonlinear multigrid solver is proposed to resolve the efficiency issue by the following approaches.The unified framework of numerical regularized moment method is first adopted to derive the high-quality discretization of the underlying problem.A fast sweeping iteration is introduced to solve the derived discrete problem more efficiently than the usual time-integration scheme on a single level grid.Taking it as the smoother,the nonlinear multigrid solver is then established to significantly improve the convergence rate.The OpenMP-based parallelization is applied in the implementation to further accelerate the computation.Numerical experiments for two lid-driven cavity flows and a bottom-heated cavity flow are carried out to investigate the performance of the resulting nonlinear multigrid solver.All results show the wonderful efficiency and robustness of the solver for both first-and second-order spatial discretization.展开更多
The structure and properties of a Keplerian rotating hyperon star with an equation of state （EOS） investigated using the relativistic σ-ω-p model are examined by employing an accurate numerical scheme. It is shown...The structure and properties of a Keplerian rotating hyperon star with an equation of state （EOS） investigated using the relativistic σ-ω-p model are examined by employing an accurate numerical scheme. It is shown that there is a clear rotating effect on the structure and properties, and that hyperon star matter cannot support a star with a mass larger than 1.9 M~, even a star rotating at the fastest allowed frequency. The constraints of the two known fastest rotating frequencies （716 Hz and 1122 Hz） on the mass and radius of a hyperon star are also explored. ~rthermore, our results indicate that the imprint of the rapid rotation of a hyperon star on the moment of inertia is clear; the backward equatorial redshift, the forward equatorial redshift and the polar redshift can be distinguished clearly, the forward equatorial redshift is always negative; and its figuration is far from a spherical symmetric shape.展开更多
The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole,and it can be decomposed into the following two functions by similarity transformation:one is a function of time...The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole,and it can be decomposed into the following two functions by similarity transformation:one is a function of time(the particle k-th moments),and the other is a function of dimensionless volume(self-preserving size distribution).In this paper,a simple iterative direct numerical simulation(iDNS)is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment,which has been solved with the Taylor-series expansion method of moment(TEMOM)in our previous work.The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature,and then the method is extended to the field of Brownian agglomeration over the entire size range.The results show that the difference between the lognormal function and the self-preserving size distribution is significant.Moreover,the thermodynamic constraint of the algebraic mean volume is also investigated.In short,the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship;thus,a complete method to solve the Smoluchowski coagulation equation asymptotically is established.展开更多
This paper discusses the pth moment stability of neutral stochastic differential equations with multiple variable delays. The equation has a much more general form than the neutral stochastic differential equations wi...This paper discusses the pth moment stability of neutral stochastic differential equations with multiple variable delays. The equation has a much more general form than the neutral stochastic differential equations with delay. A new kind of φ-function is introduced to address the stability, which is more general than the exponential stability and polynomial stability. Using a specific Lyapunov function, a stability criteria for the neutral stochastic differential equations with multiple variable delays is established, by which it is relatively easy to verify the stability of such equations. Finally, the proposed theories are illustrated by two examples.展开更多
基金supported by the National Research Foundation of Korea (NRF-2017R1C1B1005436)the TJ Park Science Fellowship of POSCO TJ Park Foundation
文摘In this article, we study the nonlinear stochastic heat equation in the spatial domain R^d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z^d to that on R^d.Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.
基金supported in part by a Simons Foundation GrantThe research of YH is supported in part by an NSERC Discovery grant and a startup fund from University of Alberta at Edmonton.
文摘This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model ∂u/∂t=1/2△+u˙W on[0,∞)×R^(d) with d≥1 has a unique randomfield solution,where W(t,x)is a fractional Brownian sheet on[0,∞)×Rd and formally ˙W=∂d+1/∂t+∂x_(1)…∂x_(d)=W(t,x).When the noise W(t,x) is white in time,our condition is both necessary and sufficient when the initial data u(0,x)is bounded between two positive constants.When the noise is fractional in time with Hurst parameter H_(0)>1/2,our sufficient condition,which improves the known results in the literature,is different from the necessary one.
基金supported in part by Natural Science Foundation of BeijingMunicipality(No.180001)National Natural Science Foundation of China(Grant No.12090022)supported by the National Natural Science Foundation of China(Grant No.12171026,U1930402 and 12031013).
文摘As one of the main governing equations in kinetic theory,the Boltzmann equation is widely utilized in aerospace,microscopic flow,etc.Its high-resolution simulation is crucial in these related areas.However,due to the high dimensionality of the Boltzmann equation,high-resolution simulations are often difficult to achieve numerically.The moment method which was first proposed in Grad(Commun Pure Appl Math 2(4):331-407,1949)is among the popular numerical methods to achieve efficient high-resolution simulations.We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation,which effectively reduces the dimensionality of the problem.However,one of themain challenges is that it leads to an unclosed moment system,and closure is needed to obtain a closedmoment system.It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory.Other than the traditional human designs of closures,the machine learning-based approach has attracted much attention lately in Han et al.(Proc Natl Acad Sci USA 116(44):21983-21991,2019)and Huang et al.(J Non-Equilib Thermodyn 46(4):355-370,2021).In this work,we propose a machine learning-based method to derive a moment closure model for the Boltzmann-BGK equation.In particular,the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances,i.e.,the Galilean invariance,reflecting invariance,and scaling invariance,inherited from the original Boltzmann-BGK equation and playing an important role in the correct simulation of the Boltzmann equation.Numerical simulations on the 1D-1D examples including the smooth and discontinuous initial condition problems,Sod shock tube problem,the shock structure problems,and the 1D-3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.
基金partially supported by the National Natural Science Foundation of China,No.12171240the Fundamental Research Funds for the Central Universities,China,No.NS2021054.
文摘We study efficient simulation of steady state for multi-dimensional rarefied gas flow,which is modeled by the Boltzmann equation with BGK-type collision term.A nonlinear multigrid solver is proposed to resolve the efficiency issue by the following approaches.The unified framework of numerical regularized moment method is first adopted to derive the high-quality discretization of the underlying problem.A fast sweeping iteration is introduced to solve the derived discrete problem more efficiently than the usual time-integration scheme on a single level grid.Taking it as the smoother,the nonlinear multigrid solver is then established to significantly improve the convergence rate.The OpenMP-based parallelization is applied in the implementation to further accelerate the computation.Numerical experiments for two lid-driven cavity flows and a bottom-heated cavity flow are carried out to investigate the performance of the resulting nonlinear multigrid solver.All results show the wonderful efficiency and robustness of the solver for both first-and second-order spatial discretization.
基金supported by the National Natural Science Foundation of China (Grant No. 10947023)the Fundamental Research Funds for the Central University,China (Grant No. 2009ZM0193)
文摘The structure and properties of a Keplerian rotating hyperon star with an equation of state （EOS） investigated using the relativistic σ-ω-p model are examined by employing an accurate numerical scheme. It is shown that there is a clear rotating effect on the structure and properties, and that hyperon star matter cannot support a star with a mass larger than 1.9 M~, even a star rotating at the fastest allowed frequency. The constraints of the two known fastest rotating frequencies （716 Hz and 1122 Hz） on the mass and radius of a hyperon star are also explored. ~rthermore, our results indicate that the imprint of the rapid rotation of a hyperon star on the moment of inertia is clear; the backward equatorial redshift, the forward equatorial redshift and the polar redshift can be distinguished clearly, the forward equatorial redshift is always negative; and its figuration is far from a spherical symmetric shape.
基金This research was funded by the National Natural Science Foundation of China with grant numbers 11972169 and 11902075.
文摘The particle number density in the Smoluchowski coagulation equation usually cannot be solved as a whole,and it can be decomposed into the following two functions by similarity transformation:one is a function of time(the particle k-th moments),and the other is a function of dimensionless volume(self-preserving size distribution).In this paper,a simple iterative direct numerical simulation(iDNS)is proposed to obtain the similarity solution of the Smoluchowski coagulation equation for Brownian motion from the asymptotic solution of the k-th order moment,which has been solved with the Taylor-series expansion method of moment(TEMOM)in our previous work.The convergence and accuracy of the numerical method are first verified by comparison with previous results about Brownian coagulation in the literature,and then the method is extended to the field of Brownian agglomeration over the entire size range.The results show that the difference between the lognormal function and the self-preserving size distribution is significant.Moreover,the thermodynamic constraint of the algebraic mean volume is also investigated.In short,the asymptotic solution of the TEMOM and the self-preserving size distribution form a one-to-one mapping relationship;thus,a complete method to solve the Smoluchowski coagulation equation asymptotically is established.
基金The National Natural Science Foundation of China (No.10671078)
文摘This paper discusses the pth moment stability of neutral stochastic differential equations with multiple variable delays. The equation has a much more general form than the neutral stochastic differential equations with delay. A new kind of φ-function is introduced to address the stability, which is more general than the exponential stability and polynomial stability. Using a specific Lyapunov function, a stability criteria for the neutral stochastic differential equations with multiple variable delays is established, by which it is relatively easy to verify the stability of such equations. Finally, the proposed theories are illustrated by two examples.
