In this paper,we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation(CNOP)approach for the EI Ni˜no-Southern Oscillation(ENSO)model.This method consists of solving the EN...In this paper,we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation(CNOP)approach for the EI Ni˜no-Southern Oscillation(ENSO)model.This method consists of solving the ENSO model by using a mixed 4th-order Runge-Kutta method.Convergence,the local and global truncation error of this mixed 4th-order Runge-Kutta method are proved.Furthermore,optimal control problem is developed and the gradient of the cost function is determined.展开更多
This work presents the mathematical framework of the “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N),” which generalizes and extends all...This work presents the mathematical framework of the “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N),” which generalizes and extends all of the previous works performed to date on this subject. The 5<sup>th</sup>-CASAM-N enables the exact and efficient computation of all sensitivities, up to and including fifth-order, of model responses to uncertain model parameters and uncertain boundaries of the system’s domain of definition, thus enabling, inter alia, the quantification of uncertainties stemming from manufacturing tolerances. The 5<sup>th</sup>-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.展开更多
In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order ...In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.展开更多
In this article, we study the phase-field model of solidification for numerical simulation of dendritic crystal growth that occurs during the casting of metals and alloys. Phase-field model of solidification describes...In this article, we study the phase-field model of solidification for numerical simulation of dendritic crystal growth that occurs during the casting of metals and alloys. Phase-field model of solidification describes the physics of dendritic growth in any material during the process of under cooling. The numerical procedure in this work is based on finite difference scheme for space and the 4th-order Runge-Kutta method for time discretization. The effect of each physical parameter on the shape and growth of dendritic crystal is studied and visualized in detail.展开更多
This paper deals with a homogeneous Neumann initial-boundary problem of a 4th-order parabolic equation modeling epitaxial growth of thin film. We determine the classification of initial energy on the existence of blow...This paper deals with a homogeneous Neumann initial-boundary problem of a 4th-order parabolic equation modeling epitaxial growth of thin film. We determine the classification of initial energy on the existence of blow-up, global existence and extinction of solutions by using the potential well method and the auxiliary function method.Moreover, asymptotic estimates on global solution and extinction solution are studied,respectively.展开更多
基金supported in part by NSF of China(No.11371031),Technology Infrastructure Work(No.2014FY210100)Baoji Science and Technology Plan Projects(No.14SFGG-2-7),and the Key Project of Baoji University of Arts and Sciences(No.ZK15033).
文摘In this paper,we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation(CNOP)approach for the EI Ni˜no-Southern Oscillation(ENSO)model.This method consists of solving the ENSO model by using a mixed 4th-order Runge-Kutta method.Convergence,the local and global truncation error of this mixed 4th-order Runge-Kutta method are proved.Furthermore,optimal control problem is developed and the gradient of the cost function is determined.
文摘This work presents the mathematical framework of the “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N),” which generalizes and extends all of the previous works performed to date on this subject. The 5<sup>th</sup>-CASAM-N enables the exact and efficient computation of all sensitivities, up to and including fifth-order, of model responses to uncertain model parameters and uncertain boundaries of the system’s domain of definition, thus enabling, inter alia, the quantification of uncertainties stemming from manufacturing tolerances. The 5<sup>th</sup>-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.
文摘In this paper we present and test a numerical method for computing eigenvalues of 4th order Sturm-Liouville (SL) differential operators on finite intervals with regular boundary conditions. This method is a 4th order shooting method based on Magnus expansions (MG4) which use MG4 shooting as the integrator. This method is similar to the SLEUTH (Sturm-Liouville Eigenvalues Using Theta Matrices) method of Greenberg and Marletta which uses the 2nd order Pruess method (also known as the MG2 shooting method) for the integrator. This method often achieves near machine precision accuracies, and some comparisons of its performance against the well-known SLEUTH software package are presented.
文摘In this article, we study the phase-field model of solidification for numerical simulation of dendritic crystal growth that occurs during the casting of metals and alloys. Phase-field model of solidification describes the physics of dendritic growth in any material during the process of under cooling. The numerical procedure in this work is based on finite difference scheme for space and the 4th-order Runge-Kutta method for time discretization. The effect of each physical parameter on the shape and growth of dendritic crystal is studied and visualized in detail.
基金Supported by Shandong Provincial Natural Science Foundation of China(Grant No.ZR2021MA003,ZR2020MA020).
文摘This paper deals with a homogeneous Neumann initial-boundary problem of a 4th-order parabolic equation modeling epitaxial growth of thin film. We determine the classification of initial energy on the existence of blow-up, global existence and extinction of solutions by using the potential well method and the auxiliary function method.Moreover, asymptotic estimates on global solution and extinction solution are studied,respectively.
