Partitioned Runge-Kutta Methods

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. 

Gunther M, Kvasmo A, Rentrop P (2001) Multirate partitioned Runge-Kutta methods. BIT 41 504-514... 

It can be seen that Runge-Kutta methods treat all components of the differential equation identically. On the other hand, in molecular dynamics the equations of motion often have a special structure for example the differential equation system is typically linear inp, and, moreover, the equations have a special coupling structure so that the differential equation for q depends only onp and that for/> depends only on q. So-called partitioned Runge-Kutta methods allow us to exploit this structure. As an illustration, consider the method ... 

The more general family of Partitioned Runge-Kutta methods is defined by making use of a partitioning of the system and introducing combinations of a set of internal stages. This more general family of schemes is discussed in some detail in [326] (see also discussions of [164,227]). 

As a special case of a partitioned Runge-Kutta method, consider the Newmark family of methods [280] defined for two parameters o and r] by the formulas... 

All Runge-Kutta methods and Partitioned Runge-Kutta methods are affine invariant, thus, if they are also symmetric, then they preserve time-reversal symmetry. 

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... 

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 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). 

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 ... 

Mur97] Murua A. (1997) Partitioned half-explicit Runge-Kutta methods for differential algebraic systems of index 2. Computing 59 43-61. 

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