We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valu...We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.展开更多
In this paper,we present quantum algorithms for a class of highly-oscillatory transport equations,which arise in semi-classical computation of surface hopping problems and other related non-adiabatic quantum dynamics,...In this paper,we present quantum algorithms for a class of highly-oscillatory transport equations,which arise in semi-classical computation of surface hopping problems and other related non-adiabatic quantum dynamics,based on the Born-Oppenheimer approximation.Our method relies on the classical nonlinear geometric optics method,and the recently developed Schrödingerisation approach for quantum simulation of partial differential equations.The Schrödingerisation technique can transform any linear ordinary and partial differential equations into Hamiltonian systems evolving under unitary dynamics,via a warped phase transformation that maps these equations to one higher dimension.We study possible paths for better recoveries of the solution to the original problem by shifting the bad eigenvalues in the Schrödingerized system.Our method ensures the uniform error estimates independent of the wave length,thus allowing numerical accuracy,in maximum norm,even without numerically resolving the physical oscillations.Various numerical experiments are performed to demonstrate the validity of this approach.展开更多
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11371218, 91630205）.
文摘We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.
文摘In this paper,we present quantum algorithms for a class of highly-oscillatory transport equations,which arise in semi-classical computation of surface hopping problems and other related non-adiabatic quantum dynamics,based on the Born-Oppenheimer approximation.Our method relies on the classical nonlinear geometric optics method,and the recently developed Schrödingerisation approach for quantum simulation of partial differential equations.The Schrödingerisation technique can transform any linear ordinary and partial differential equations into Hamiltonian systems evolving under unitary dynamics,via a warped phase transformation that maps these equations to one higher dimension.We study possible paths for better recoveries of the solution to the original problem by shifting the bad eigenvalues in the Schrödingerized system.Our method ensures the uniform error estimates independent of the wave length,thus allowing numerical accuracy,in maximum norm,even without numerically resolving the physical oscillations.Various numerical experiments are performed to demonstrate the validity of this approach.
