J. M. Franco

An embedded pair of exponentially fitted explicit Runge-Kutta methods

An embedded pair of exponentially fitted explicit Runge-Kutta (RK) methods for the numerical integration of IVPs with oscillatory solutions is derived. This pair is based on the exponentially fitted explicit RK method constructed in Vanden Berghe et al., and we confirm that the methods which constitute the pair have algebraic order 4 and 3. Some numerical experiments show the efficiency of our pair when it is compared with the variable step code proposed by Vanden Berghe et al. (J. Comput. Appl. Math. 125 (2000) 107).

Exponentially fitted symplectic integrators of RKN type for solving oscillatory problems

Abstract A class of explicit modified Runge–Kutta–Nystrom (RKN) methods for the numerical integration of second-order IVPs with oscillatory solutions is presented. The symplecticness conditions and the exponential fitting conditions for this class of methods are derived. Based on this conditions, explicit modified RKN integrators with two and three stages per step which have algebraic orders two and four, respectively, are constructed. These new integrators preserve symplecticness when they are applied to Hamiltonian problems, and they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions { exp ( λ t ) , exp ( − λ t ) } , λ ∈ C , or equivalently { sin ( ω t ) , cos ( ω t ) } when λ = i ω , ω ∈ R . We also analyze the stability properties of the new integrators, obtaining generalized periodicity regions for the classical second-order linear test model. The numerical experiments carried out show that the new methods are more efficient than other symplectic and exponentially fitted codes proposed in the scientific literature.

Minimum storage Runge-Kutta schemes for computational acoustics

Abstract In this paper, an implementation of some Runge-Kutta schemes that requires 2N-storage, where N is the number of degrees of freedom of the system is proposed. It is shown that, some Runge-Kutta schemes for wave propagation proposed by Hu, Hussaini and Manthey [1] can be written by using our implementation. Thus, we provide an alternative to the 2N-storage implementation proposed by Stanescu and Habashi [2] which follows the approach of Williamson [3], Carpenter and Kennedy [4] and others. In addition, new Runge-Kutta schemes with small dissipation and dispersion errors that can be written also by using our implementation are given. Finally, the results of some numerical experiments are presented to compare the behaviour of these methods.

Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order

New SDIRKN methods specially adapted to the numerical integration of second-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion and P-stability conditions for SDIRKN methods which are assumed to be zero dissipative. Two four-stage symplectic and P-stable methods with algebraic order 4 and high order of dispersion are obtained. One of the methods is symmetric and sixth-order dispersive whereas the other method is nonsymmetric and eighth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with other methods derived by Sharp et al. [IMA J. Numer. Anal. 10 (1990) 489–504].

Explicit Runge-Kutta methods for initial value problems with oscillating solutions

New pairs of embedded Runge-Kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained. These pairs have been derived taking into account not only the usual properties of accuracy, stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by Van der Houwen and Sommeijer (1989). Three nine-stage embedded pairs of Runge-Kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and/or dissipation are selected among the members of a family of pairs depending on several free parameters. Some numerical results are presented to show the efficiency of the new methods.

Sixth-order symmetric and symplectic exponentially fitted modified Runge-Kutta methods of Gauss type

Abstract The construction of symmetric and symplectic exponentially fitted modified Runge–Kutta (RK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is considered. In a previous paper [H. Van de Vyver, A fourth order symplectic exponentially fitted integrator, Comput. Phys. Comm. 176 (2006) 255–262] a two-stage fourth-order symplectic exponentially fitted modified RK method has been proposed. Here, two three-stage symmetric and symplectic exponentially fitted integrators of Gauss type, either with fixed nodes or variable nodes, are derived. The algebraic order of the new integrators is also analyzed, obtaining that they possess sixth-order as the classical three-stage RK Gauss method. Numerical experiments with some oscillatory problems are presented to show that the new methods are more efficient than other symplectic RK Gauss codes proposed in the scientific literature.

SDIRK methods for stiff ODEs with oscillating solutions

New SDIRK methods specially adapted to the numerical solution of stiff systems of ODEs which are assumed to possess oscillating solutions are obtained. Our interest is centered on the dispersion (phase errors) and the dissipation (numerical damping), of the dominant components in the numerical oscillations when these methods are applied to a homogeneous linear test model. Two A-stable methods with algebraic order 3 and higher order of dissipation are obtained among the members of a family of methods proposed by Van der Houwen and Sommeijer (1989). Some numerical results are presented to show the efficiency of the new methods when they are compared with other methods presented in Van der Houwen and Sommeijer (1989).

Symmetric and symplectic exponentially fitted Runge–Kutta methods of high order

Abstract The construction of high order symmetric, symplectic and exponentially fitted Runge–Kutta (RK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is analyzed. Based on the symplecticness, symmetry, and exponential fitting properties, three new four-stage RK integrators, either with fixed- or variable-nodes, are constructed. The algebraic order of the new integrators is also studied, showing that they possess eighth-order of accuracy as the classical four-stage RK Gauss method. Numerical experiments with some oscillatory test problems are presented to show that the new methods are more efficient than other symplectic four-stage eighth-order RK Gauss codes proposed in the scientific literature.

Symplectic explicit methods of Runge-Kutta-Nyström type for solving perturbed oscillators

The construction of symplectic methods of Runge-Kutta-Nystrom type (RKN-type) specially adapted to the numerical solution of perturbed oscillators is analyzed. Based on the symplecticity conditions for this class of methods, new fourth-order explicit methods of RKN type for solving perturbed oscillators are constructed. The derivation of the new symplectic methods is carried out paying special attention to the minimization of the principal term of the local truncation error. The numerical experiments carried out show the qualitative behavior and the efficiency of the new methods when they are compared with some standard and specially adapted symplectic methods proposed in the scientific literature for solving second-order oscillatory differential systems.

Fourth-Order Symmetric DIRK Methods for Periodic Stiff Problems

New symmetric DIRK methods specially adapted to the numerical integration of first-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion conditions for symmetric DIRK methods as well as symmetric stability functions with real poles and maximal dispersion order. Two new fourth-order symmetric methods with four and five stages are obtained. One of the methods is fourth-order dispersive whereas the other method is symplectic and sixth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with the symplectic DIRK method derived by Sanz-Serna and Abia (SIAM J. Numer. Anal. 28 (1991) 1081–1096).