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Journal ArticleDOI

Construction of Exponentially Fitted Symplectic Runge–Kutta–Nyström Methods from Partitioned Runge–Kutta Methods

01 Aug 2016-Mediterranean Journal of Mathematics (Springer International Publishing)-Vol. 13, Iss: 4, pp 2271-2285
TL;DR: This work constructs RKN methods from PRK methods with up to five stages and fourth algebraic order, andumerical results are given for the two-body problem and the perturbed two- body problem.
Abstract: In this work, we give the general framework for constructing trigonometrically fitted symplectic Runge–Kutta–Nystrom (RKN) methods from symplectic trigonometrically fitted partitioned Runge–Kutta (PRK) methods. We construct RKN methods from PRK methods with up to five stages and fourth algebraic order. Numerical results are given for the two-body problem and the perturbed two-body problem.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients is considered, and a hybrid Numerov method is used that is constructed in the sense of Runge-Kutta ones.
Abstract: In this paper, we consider the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method.

81 citations


Cites methods from "Construction of Exponentially Fitte..."

  • ...In parallel, our group has dealt in-depth with various types of multistep methods, Runge-Kutta & RKN methods, and especially fitted methods for addressing Equation 2, see previous studies.(16-43) Here, we will extend the results of Tsitouras(14) introducing an eighth algebraic order method at a cost of 5 stages per step....

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Journal ArticleDOI
TL;DR: In this paper, a three-stages two-stage method was proposed to solve the Schrodinger problem in high algebraic order, where the approximation of the first layer is done on the point of the point $$x n-1} and not on the usual point.
Abstract: In this paper we introduce, for the first time in the literature, a three-stages two-step method. The new algorithm has the following characteristics: (1) it is a two-step algorithm, (2) it is a symmetric method, (3) it is an eight-algebraic order method (i.e of high algebraic order), (4) it is a three-stages method, (5) the approximation of its first layer is done on the point $$x_{n-1}$$ and not on the usual point $$x_{n}$$ , (6) it has eliminated the phase–lag and its derivatives up to order two, (7) it has good stability properties (i.e. interval of periodicity equal to $$\left( 0, 22 \right) $$ . For this method we present a detailed analysis : development, errorand stability analysis. The new proposed algorithm is applied to systems of differential equations of the Schrodinger type in order to examine its efficiency.

74 citations

Journal ArticleDOI
TL;DR: In this article, the integration of the special second-order initial value problem is considered and a new family of hybrid Numerov methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step.
Abstract: In this paper, we consider the integration of the special second-order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase-lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.

67 citations

Journal ArticleDOI
TL;DR: In this paper, a new finite diffence pair is presented, which is of symmetric two-step, is four-stages, is of tenth-algebraic order, and has excellent stability properties for all type of problems.
Abstract: A new finite difference pair is produced in this paper, for the first time in the literature. The characteristics of the new finite diffence pair are: (1) is of symmetric two-step, (2) is four-stages, (3) is of tenth-algebraic order, (4) the production of the pair is based on the following approximations for the layers: first and second layer are approximated on the point $$x_{n-1}$$ , third layer is approximated on the point $$x_{n}$$ and finally fourth layer is approximated on the point $$x_{n+1}$$ , (5) has vanished the phase-lag and its first and second derivatives, (6) has excellent stability properties for all type of problems, (7) has an interval of periodicity equal to $$\left( 0, \infty \right) $$ . We present for the new obtained finite difference pair a full theoretical analysis. The effectiveness of the new developed finite difference pair is proved by its application on systems of coupled differential equations arising from the Schrodinger equation.

65 citations

References
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Book
01 Jan 1994
TL;DR: Examples of Hamiltonian Systems, symplectic integration, and Numerical Methods: Checking preservation of area: Jacobians, and Necessity of the symplecticness conditions.
Abstract: Hamiltonian Systems. Examples of Hamiltonian Systems. Symplecticness. The solution operator. Preservation of area. Checking preservation of area: Jacobians. Checking preservation of area: differential forms. Symplectic transformations. Conservation of volume. Numerical Methods. Numerical integrators. Stiff problems. Runge-Kutta methods. Partitioned Runge-Kutta methods. Runge-Kutta-Nystrom methods. Composition of methods - adjoints. Order conditions. Order conditions for Runge-Kutta methods. The local error in Runge-Kutta methods. Order conditions for PRK methods. The local error in Partitioned Runge-Kutta methods. Order conditions for Runge-Kutta-Nystrom methods. The local error in Runge-Kutta-Nystrom mehthods. Implementation. Variable step sizes. Embedded pairs. Numerical experience with variable step sizes. Implementing implicit methods. Fourth-order Gauss method. Symplectic integration. Symplectic methods. Symplectic Runge-Kutta methods. Symplectic partitioned Runge-Kutta methods. Symplectic Runge-Kutta-Nystrom methods. Necessity of the symplecticness conditions. Symplectic order conditions. Prelimiaries. Order conditions for symplectic RK methods. Order conditions for symplectic PRK methods. Order conditions for symplectic RKN methods. Homogenous form of the order conditions. Available symplectic methods. Symplecticness of the Gauss methods. Diagonally implicity Runge-Kutta methods. Other symplectic Runge-Kutta methods. Explicit partitioned Runge-Kutta methods. Available symplectic Runge-Kutta-Nystrom methods. Numerical experiments. A comparison of symplectic integrators. Variable step sizes for symplectic methods. Conclusions and recommendations. Properties of symplectic integrators. Backward error interpretation. An alternative approach. Conservation of energy. KAM theory. Generating functions. The concept of generating function. Hamilton-Jacobi equations. Integrators based on generating functions. Generating functions for RK methods. Canonical order theory. Lie formalism. The Poisson bracket. Lie operators and Lie series. The Baker-Campbell-Hausdorff formula. Application to fractional-step methods. Extension to the non-Hamilton case. High-order methods. High-order Lie methods. High-order Runge-Kutta-Nystrom methods. A comparison of order 8 symplectic integrators. Extensions. Partitioned Runge-Kutta methods for nonseparable Hamiltonian systems. Canonical B-series. Conjugate symplectic methods. Trapezoidal rule. Constrained systems. General Poisson structures. Multistep methods. Partial differential equations. Reversable systems. Volume preserving flows.

1,327 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed an explicit third order symplectic map (i.e., a third order integration step which preserves exactly the canonical character of the equations of motion) and indicated the method for higher order maps.
Abstract: The class of differential equations of interest to this paper is that in which the equations are derivable from a Hamiltonian by the use of Hamilton's equations. The exact solution of such a system of differential equations leads to a symplectic map from the initial conditions to the present state of the system. A characteristic feature of all explicit higher order integration methods, however, is that they are not exactly symplectic. In many applications the salient features appear only after a long time or after numerous iterations; in these applications spurious damping or excitation may lead to misleading results. The purpose of this paper is to develop an explicit third order symplectic map (i.e. a third order integration step which preserves exactly the canonical character of the equations of motion) and to indicate the method for higher order maps. For a typical numerical integration, this method can be used to eliminate the noncanonical effects while providing the accuracy corresponding to a third order integration step.

696 citations

Journal ArticleDOI
01 Jan 1983
TL;DR: In this paper, the authors developed an explicit third order symplectic map (i.e., a third order integration step which preserves exactly the canonical character of the equations of motion) and indicated the method for higher order maps.
Abstract: The class of differential equations of interest to this paper is that in which the equations are derivable from a Hamiltonian by the use of Hamilton's equations. The exact solution of such a system of differential equations leads to a symplectic map from the initial conditions to the present state of the system. A characteristic feature of all explicit higher order integration methods, however, is that they are not exactly symplectic. In many applications the salient features appear only after a long time or after numerous iterations; in these applications spurious damping or excitation may lead to misleading results. The purpose of this paper is to develop an explicit third order symplectic map (i.e. a third order integration step which preserves exactly the canonical character of the equations of motion) and to indicate the method for higher order maps. For a typical numerical integration, this method can be used to eliminate the noncanonical effects while providing the accuracy corresponding to a third order integration step.

463 citations

Journal ArticleDOI
TL;DR: Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order, and a new, more accurate way of applying the methods thus obtained to compositions of an arbitrary first-order integrator is described and tested.
Abstract: Differential equations of the form $\dot x = X = A + B$ are considered, where the vector fields A and B can be integrated exactly, enabling numerical integration of X by composition of the flows of A and B. Various symmetric compositions are investigated for order, complexity, and reversibility. Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order. A new, more accurate way of applying the methods thus obtained to compositions of an arbitrary first-order integrator is described and tested. The determining equations are explored, and new methods up to 100 times more accurate (at constant work) than those previously known are given.

378 citations

Journal ArticleDOI
TL;DR: An exponentially-fitted Runge-Kutta method for the numerical integration of initial-vakue problems with periodic or oscillating solutions with efficiency results obtained show the efficiency of the new method.

216 citations