In this paper, We show for isentropic equations of gas dynamics with adiabatic exponent gamma=3 that approximations of weak solutions generated by large time step Godunov's scheme or Glimm's scheme give entrop...In this paper, We show for isentropic equations of gas dynamics with adiabatic exponent gamma=3 that approximations of weak solutions generated by large time step Godunov's scheme or Glimm's scheme give entropy solution in the limit if Courant number is less than or equal to 1.展开更多
A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of ...A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.展开更多
As a widely used flood energy dissipator, the stepped spillway can significantly dissipate the kinetic or hydraulic energy due to the air-entrainment in skimming flow over the steps. The free-surface aeration involves...As a widely used flood energy dissipator, the stepped spillway can significantly dissipate the kinetic or hydraulic energy due to the air-entrainment in skimming flow over the steps. The free-surface aeration involves the sharp deformation of the free surface and the complex turbulent shear flows. In this study, the volume of fluid （VOF）, mixture, and Eulerian methods are utilized to simulate the air-entrainment by coupling with the Reynolds-averaged Navier-Stokes/large eddy simulation （RANS/LES） turbulence models. The free surface deformation, air volume fraction, pressure, and velocity are compared for the three different numerical methods. Only the Eulerian＋RANS method fails to capture the free-surface aeration. The air volume fraction predicted by the VOF＋LES method best matches the experimental measurement, while the mixture＋LES method predicts the inception point of the air entrainment more accurately.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
This study presents a novel neural network architecture called spectral integrated neural networks(SINNs),which combines physics-informed neural networks(PINNs)with time-spectral integration techniques to efficiently ...This study presents a novel neural network architecture called spectral integrated neural networks(SINNs),which combines physics-informed neural networks(PINNs)with time-spectral integration techniques to efficiently solve two-and three-dimensional dynamic piezoelectric problems.To avoid the numerical instability associated with time-differential operators,the coupled system of mechanical and electrical equilibrium equations is reformulated into a weak time-integral form.The temporal derivatives of displacement and voltage fields,treated as the primary unknown physical quantities,can be approximated utilizing fully connected neural networks(FCNNs).The displacements and electric potential are subsequently recovered through time-spectral integration of their respective derivatives.A physical-informed loss function is formulated by the weak time-integral type of the governing equations and boundary conditions,with the initial conditions embedded within the integral expressions.The proposed SINNs demonstrate superior stability and accuracy,even under large time steps conditions.Numerical verification is accomplished through three representative test cases of the method,and a comparison analysis is presented between the results obtained by the SINNs and those from the PINNs.展开更多
An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the...An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.展开更多
基金Supported in part by the National Natural Science of China, NSF Grant No. DMS-8657319.
文摘In this paper, We show for isentropic equations of gas dynamics with adiabatic exponent gamma=3 that approximations of weak solutions generated by large time step Godunov's scheme or Glimm's scheme give entropy solution in the limit if Courant number is less than or equal to 1.
基金The Project Supported by National Natural Science Foundation of China.
文摘A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.
基金supported by the Guangdong Special Research Fund of Public Welfare and Capacity Building(2015A020216008)the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund(the second phase)
文摘As a widely used flood energy dissipator, the stepped spillway can significantly dissipate the kinetic or hydraulic energy due to the air-entrainment in skimming flow over the steps. The free-surface aeration involves the sharp deformation of the free surface and the complex turbulent shear flows. In this study, the volume of fluid （VOF）, mixture, and Eulerian methods are utilized to simulate the air-entrainment by coupling with the Reynolds-averaged Navier-Stokes/large eddy simulation （RANS/LES） turbulence models. The free surface deformation, air volume fraction, pressure, and velocity are compared for the three different numerical methods. Only the Eulerian＋RANS method fails to capture the free-surface aeration. The air volume fraction predicted by the VOF＋LES method best matches the experimental measurement, while the mixture＋LES method predicts the inception point of the air entrainment more accurately.
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funded by the Natural Science Foundation of Shandong Province of China,Grant/Award Numbers:ZR2022YQ06,ZR2024MA002Development Plan of Youth Innovation Team in Colleges and Universities of Shandong Province,Grant/Award Number:2022KJ140+1 种基金National Natural Science Foundation of China,Grant/Award Numbers:12372199,12422207Key Laboratory of Road Construction Technology and Equipment,Grant/Award Number:300102253502.
文摘This study presents a novel neural network architecture called spectral integrated neural networks(SINNs),which combines physics-informed neural networks(PINNs)with time-spectral integration techniques to efficiently solve two-and three-dimensional dynamic piezoelectric problems.To avoid the numerical instability associated with time-differential operators,the coupled system of mechanical and electrical equilibrium equations is reformulated into a weak time-integral form.The temporal derivatives of displacement and voltage fields,treated as the primary unknown physical quantities,can be approximated utilizing fully connected neural networks(FCNNs).The displacements and electric potential are subsequently recovered through time-spectral integration of their respective derivatives.A physical-informed loss function is formulated by the weak time-integral type of the governing equations and boundary conditions,with the initial conditions embedded within the integral expressions.The proposed SINNs demonstrate superior stability and accuracy,even under large time steps conditions.Numerical verification is accomplished through three representative test cases of the method,and a comparison analysis is presented between the results obtained by the SINNs and those from the PINNs.
文摘An efficient and accurate exponential wave integrator Fourier pseudospectral （EWI-FP） method is proposed and analyzed for solving the symmetric regularized-long-wave （SRLW） equation, which is used for modeling the weakly nonlinear ion acoustic and space-charge waves. The numerical method here is based on a Gautschi-type exponential wave integrator for temporal approximation and the Fourier pseudospectral method for spatial discretization. The scheme is fully explicit and efficient due to the fast Fourier transform. Numerical analysis of the proposed EWI-FP method is carried out and rigorous error estimates are established without CFL-type condition by means of the mathematical induction. The error bound shows that EWI-FP has second order accuracy in time and spectral accuracy in space. Numerical results are reported to confirm the theoretical studies and indicate that the error bound here is optimal.
