A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. T...A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. The optimization problem related to the determination of nonlinear singular vectors and singular values is formulated. The general idea of this approach is demonstrated by a simple two-dimensional quasigeostrophic model in the atmospheric and oceanic sciences. The advantage and its applications of the new method to the predictability, ensemble forecast and finite-time nonlinear instability are discussed. This paper makes a necessary preparation for further theoretical and numerical investigations.展开更多
Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional q...Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional quasigeostrophic model in this paper. The difference between the linear and nonlinear fastest growing perturbations is demonstrated. Moreover, local nonlinear fastest growing perturbations are also found numerically. This is one of the essential differences between linear and nonlinear theories, since in former case there is no local fastest growing perturbation. The results show that the nonlinear local fastest growing perturbations play a more important role in the study of the first kind of predictability than the nonlinear global fastest growing perturbation.展开更多
The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from t...The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.展开更多
In this paper, a class of strongly nonlinear singular perturbed boundary value problems are coasidered by the theory of differential inequalities and the correction of boundary layer, under which the existence of solu...In this paper, a class of strongly nonlinear singular perturbed boundary value problems are coasidered by the theory of differential inequalities and the correction of boundary layer, under which the existence of solution is proved and the uniformly valid asymptotic expansions is obtained as well.展开更多
文摘A novel concept of nonlinear singular vector and nonlinear singular value is introduced, which is a natural generalization of the classical linear singular vector and linear singular value to the nonlinear category. The optimization problem related to the determination of nonlinear singular vectors and singular values is formulated. The general idea of this approach is demonstrated by a simple two-dimensional quasigeostrophic model in the atmospheric and oceanic sciences. The advantage and its applications of the new method to the predictability, ensemble forecast and finite-time nonlinear instability are discussed. This paper makes a necessary preparation for further theoretical and numerical investigations.
基金the National Key Basic Research Project, "Research on the FormationMechanism and Prediction Theory of Severe Synoptic Disasters in China" (Grand No. G1998040910), the National Natural Science Foundation of China (Grand Nos. 49775262 and 49823002) and t
文摘Nonlinear fastest growing perturbation, which is related to the nonlinear singular vector and nonlinear singular value proposed by the first author recently, is obtained by numerical approach for the two-dimensional quasigeostrophic model in this paper. The difference between the linear and nonlinear fastest growing perturbations is demonstrated. Moreover, local nonlinear fastest growing perturbations are also found numerically. This is one of the essential differences between linear and nonlinear theories, since in former case there is no local fastest growing perturbation. The results show that the nonlinear local fastest growing perturbations play a more important role in the study of the first kind of predictability than the nonlinear global fastest growing perturbation.
基金the National Natural Science Foundation of China(Grant Nos.10471100,40437017,and 60573158)Beijing Jiaotong University Science and Technology Foundation
文摘The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.
基金Supported by the Natural Science Foundation of Zhejiang Provivce (102009)Supported by the Natural Foundation of Huzhou Teacher's College(200302)
文摘In this paper, a class of strongly nonlinear singular perturbed boundary value problems are coasidered by the theory of differential inequalities and the correction of boundary layer, under which the existence of solution is proved and the uniformly valid asymptotic expansions is obtained as well.
