Numerical multistep methods for the efficient solution of quantum mechanics and related problems

High order closed Newton-Cotes exponentially and trigonometrically fitted formulae as multilayer symplectic integrators and their application to the radial Schrödinger equation

“…We note here that recently [107] an exponentially-fitted method for the time dependent Schrödinger equation was obtained. -Several pseudospectral methods have been studied and developed [108] -New function fitting methods [109] -Review papers have been written and Special Issues have been published in [110][111][112][113][114][115][116][117][118] 3 Basic theory on symplectic schemes and numerical methods…”

Optimizing a class of linear multi-step methods for the approximate solution of the radial Schrödinger equation and related problems with respect to phase-lag

“…• In [1,3,19] the authors present detailed reviews of the current research on the subject of this paper.…”

On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

“…For example, we may consider time-reversal symmetry of the differential flow. In such case the numerical flow maps of y = f (t, y) at t 0 and t 1 , ψ 0 h,f and ψ 1 h,f respectively, are required to satisfy ψ 1 −h,f • ψ 0 h,f (u 0 ) = u 0 for all u 0 . In addition, for the numerical integration of problems which are known to possess oscillatory solutions with some prior knowledge of its frequency, a natural requirement on the available parameters of (1.2) is that the method integrates exactly trigonometrical polynomials with this frequency.…”

“…, c q the Gauss nodes in the interval [0, 1] we have the super convergent methods with maximum order p = 2q which are both symplectic and symmetric (see [5], p. 28 and p. 179). Methods that are exact for other spaces of functions F such as trigonometrical polynomials, exponential functions or mixed algebraic and trigonometrical polynomials suitable for special types of differential equations have been considered in [1,3,4,6,7,10,11,[13][14][15][16][17][18][19][20][21].…”

“…It is interesting to remark that, as a consequence of the linearity of L j and the fact that L j [1] = 0, if F is the space of solutions of a linear homogeneous equation…”