Symplectic Runge-Kutta method

Fig. 2. The time evolution of the total energy of four water molecules (potential-energy details are given in [48]) as propagated by the symplectic Verlet method (solid) and the nonsymplectic fourth-order Runge-Kutta method (dashed pattern) for Newtonian dynamics at two timestep values.

Th. Monovasilis, Z. Kalogiratou and T. E. Simos, Computation of the eigenvalues of the Schrodinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge-Kutta methods. Physics Letters A, 2008, 372, 569-573. 

Z. Kalogiratou, Symplectic trigonometrically fitted partinioned Runge-Kutta methods. Physics Letters A, 2007, 370, 1—7. 

The behavior of the Runge-Kutta-Nystrom Symplectic method of algebraic order four developed by Sanz-Serna and Calvo12 and the behavior of the classical partitioned multistep method is similar. These methods are much more efficient that the embedded Runge-Kutta method of Dormand and Prince 5(4) (see 13). 

Let us emphasize that, while a typical RK method is not symplectic, some implicit Runge-Kutta methods are symplectic. The precise condition that must be... 

Among explicit symplectic Partitioned Runge-Kutta methods this is the maximum stability threshold [74]. In a similar way one can analyze the stability of the Verlet and other methods and one thus obtains conditions on the stepsize that must hold for the equilibrium points to be stable in the linearization. Analyzing the stability of both continuous and discrete iteration is much more compUcated for... 

The same property holds for all symplectic Runge-Kutta methods.)... 

Reich, S. On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems. Numer. Math. 76, 231-247 (1997). doi 10.1007/ s002110050261... 

In [167] the authors obtained symplectic partitioned Runge-Kutta methods (SPRK) of algebraic order two and three with phase-lag of order five. More specifically they considered systems with separable Hamiltonians of the form... 

The approximate solution of Separable Hamiltonian Systems can be done via SPRK methods. The authors used SPRK methods since there exist explicit methods of this type while SRK (symplectic Runge-Kutta) methods cannot be explicit. A PRK method can be written using the following two tableaux... 

The IE and IM methods described above turn out to be quite special in that IE s damping is extreme and IM s resonance patterns are quite severe relative to related symplectic methods. However, success was not much greater with a symplectic implicit Runge-Kutta integrator examined by Janezic and coworkers [40],... 

Thus we find that the choice of quaternion variables introduces barriers to efficient symplectic-reversible discretization, typically forcing us to use some off-the-shelf explicit numerical integrator for general systems such as a Runge-Kutta or predictor-corrector method. 

In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. 

In 52 the author develops a symplectic exponentially fitted modified Runge-Kutta-Nystrom method. The method of development was based on the development of symplectic exponentially fitted modified Runge-Kutta-Ny-strom method by Simos and Vigo-Aguiar. The new method is a two-stage second-order method with FSAL-property (first step as last). 

Symplectic integrators may be constructed in several ways. First, we may look within standard classes of methods such as the family of Runge-Kutta schemes to see if there are choices of coefficients which make the methods automatically conserve the symplectic 2-form. A second, more direct approach is based on splitting. The idea of splitting methods, often referred to in the literature as Lie-Trotter methods, is that we divide the Hamiltonian into parts, and determine the flow maps (or, in some cases, approximate flow maps) for the parts, then compose the maps to define numerical methods for the whole system. 

In ref. 153 the authors consider a family of trigonometrically fitted partitioned Runge Kutta symplectic methods of fourth order with six stages. The radial time-independent Schrodinger equation may be written in the form ... 

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