The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics.We demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear...The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics.We demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear system,whereby each diagonal block is a high frequency Helmholtz problem.Based on this model,we derive indefinite Helmholtz model problems with strongly varying wavenumbers.We employ the iterative approach for their solution.In particular,we develop a preconditioner that has its spectrum restricted to a quadrant(of the complex plane)thereby making it easily invertible by multigrid methods with standard components.This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems.The aim of this study is to report the feasibility of this preconditioner for the model problems.We compare this idea with the other prevalent preconditioning ideas,and discuss its merits.Results of numerical experiments are presented,which complement the proposed ideas,and show that this preconditioner may be used in an automatic setting.展开更多
In this study,we discuss the central force problem by using the nonlocal-in-time kinetic energy approach.At low length scales,the system is dominated by the generalized 4^(th)-order extended Fisher-Kolmogorov stationa...In this study,we discuss the central force problem by using the nonlocal-in-time kinetic energy approach.At low length scales,the system is dominated by the generalized 4^(th)-order extended Fisher-Kolmogorov stationary equation and by the 4^(th)-order stationary Swift-Hohenberg differential equation under explicit conditions.The energy is a conserved quantity along orbits of the extended Fisher-Kolmogorov stationary equation.The system is quantized,the system is stable,and the ground energy problem is solved.展开更多
基金funded partially by Fonds voor Wetenschappelijk Onderzoek(FWO Bel-gium)projects G.0174.08 and 1.5.145.10,by the University of Antwerp,Belgium,and by the Institute of Business Administration,Karachi,Pakistan.We wish to thank the sponsors sincerely for their support.
文摘The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics.We demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear system,whereby each diagonal block is a high frequency Helmholtz problem.Based on this model,we derive indefinite Helmholtz model problems with strongly varying wavenumbers.We employ the iterative approach for their solution.In particular,we develop a preconditioner that has its spectrum restricted to a quadrant(of the complex plane)thereby making it easily invertible by multigrid methods with standard components.This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems.The aim of this study is to report the feasibility of this preconditioner for the model problems.We compare this idea with the other prevalent preconditioning ideas,and discuss its merits.Results of numerical experiments are presented,which complement the proposed ideas,and show that this preconditioner may be used in an automatic setting.
文摘In this study,we discuss the central force problem by using the nonlocal-in-time kinetic energy approach.At low length scales,the system is dominated by the generalized 4^(th)-order extended Fisher-Kolmogorov stationary equation and by the 4^(th)-order stationary Swift-Hohenberg differential equation under explicit conditions.The energy is a conserved quantity along orbits of the extended Fisher-Kolmogorov stationary equation.The system is quantized,the system is stable,and the ground energy problem is solved.
