Symmetric Multistep Methods Research Articles

Multistep cosine methods for second-order partial differential systems

In this paper we construct and analyse some multistep methods that integrate exactly the stiff part of a second-order partial differential equation. Much emphasis is given to symmetric methods of this type in order to deal with Hamiltonian problems. In such a way, we obtain very efficient methods because they are explicit, stable and, in the case of symmetric methods, conserve properties of the system that they imitate, as is shown in a forthcoming paper. The Gautschi method is the simplest of these methods. We analyse it here using different techniques and assumptions from those in the literature, which also allow the study of methods of higher order. In particular, a symmetric fourth-order multistep method of this type is thoroughly constructed and analysed, including its resonances and possible ways to filter them.

Symmetric multistep methods with zero phase-lag for periodic initial value problems of second order differential equations

We present in this paper two-step and four-step symmetric multistep methods involving a parameter p to solve a special class of initial value problems associated with second order ordinary differential equations in which the first derivative does not appear explicitly. It is shown that the methods have zero phase-lag when p is chosen as 2π times the frequency of the given initial value problem. The periodicity intervals are given in terms of expressions involving the parameter p. As p increases, the periodicity intervals increase and for large p, the methods are almost P-stable.

Dissipative trigonometrically fitted methods for the numerical solution of orbital problems

In this paper a new approach for the numerical solution of orbital problems is introduced. This approach is based on the dissipative multistep methods (i.e., methods which are not symmetric). We show that the trigonometrical fitting property produces efficient methods of this category. For comparison purposes we use well-known symmetric multistep methods.

Trigonometrically-fitted symmetric multistep methods for the approximate solution of orbital problems

An explicit hybrid symmetric eight-step method of algebraic order ten is presented in this paper. Numerical comparative results from the application of the new method to well-known periodic orbital problems, clearly demonstrate the superior efficiency of the method presented in this paper.

An exponentially-fitted and trigonometrically-fitted method for the numerical solution of periodic initial-value problems

An explicit symmetric multistep method is presented in this paper. The new method is exponentially fitted and trigonometrically-fitted and is of algebraic order eight. The effectiveness of the exponential fitting is proved by the application of the new method and the classical one (with constant coefficients) to well-known periodic problems.

Family of Twelve Steps Exponential Fitting Symmetric Multistep Methods for the Numerical Solution of the Schrödinger Equation

In the present paper we present a family of twelve steps symmetric multistep methods. The explicit part of new family of methods is applied to the scattering problems of the radial Schrodinger equation. This application shows the efficiency of the new family of methods.

An Exponentially Fitted and Trigonometrically Fitted Method for the Numerical Solution of Orbital Problems

An explicit symmetric multistep method is presented. The new method is exponentially fitted and trigonometrically fitted and is of algebraic order 10. The efficiency of the proposed method is demonstrated by its application to well-known periodic orbital problems.

Long-Term Integration Error of Kustaanheimo-Stiefel Regularized Orbital Motion

We confirm that the positional error of a perturbed two-body problem expressed in the Kustaanheimo-Stiefel (K-S) variable is proportional to the fictitious time s, which is the independent variable in the K-S transformation. This property does not depend on the type of perturbation, on the integrator used, or on the initial conditions, including the nominal eccentricity. The error growth of the physical time evolution and the Kepler energy is proportional to s if the perturbed harmonic oscillator part of the equation of motion is integrated by a time-symmetric integration formula, such as the leapfrog or the symmetric multistep method, and is proportional to s2 when using traditional integrators, such as the Runge-Kutta, Adams, Stormer, and extrapolation methods. Also, we discovered that the K-S regularization avoids the step size resonance/instability of the symmetric multistep method that appears in the unregularized cases. Therefore, the K-S regularized equation of motion is useful to investigate the long-term behavior of perturbed two-body problems, namely, those used for studying the dynamics of comets, minor planets, the Moon, and other natural and artificial satellites.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

Parallel/Vector Integration Methods for Dynamical Astronomy

This paper reviews three recent1 works on the numerical methods to integrate ordinary differential equations (ODE), which are specially designed for parallel, vector, and/or multi-processor-unit (PU) computers. The first is the Picard-Chebyshev method (Fukushima, 1997a). It obtains a global solution of ODE in the form of Chebyshev polynomial of large (> 1000) degree by applying the Picard iteration repeatedly. The iteration converges for smooth problems and/or perturbed dynamics. The method runs around 100–1000 times faster in the vector mode than in the scalar mode of a certain computer with vector processors (Fukushima, 1997b). The second is a parallelization of a symplectic integrator (Sana et al., 1997). It regards the implicit midpoint rules covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method is applicable to Hamiltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with more than 1000 PUs. The last is a parallelization of the extrapolation method (Ito and Fukushima, 1997). It performs trial integrations in parallel. Also the trial integrations are further accelerated by balancing computational load among PUs by the technique of folding. The method is all-purpose and achieves an acceleration factor of around 3.5 by using several PUs. Finally, we give a perspective on the parallelization of some implicit integrators which require multiple corrections in solving implicit formulas like the implicit Hermitian integrators (Makino and Aarseth, 1992), (Hut et al., 1995) or the implicit symmetric multistep methods (Fukushima, 1998), (Fukushima, 1999).

Parallel/Vector Integration Methods for Dynamical Astronomy

AbstractThis paper reviews three recent works on the numerical methods to integrate ordinary differential equations (ODE), which are specially designed for parallel, vector, and/or multi-processor-unit (PU) computers. The first is the Picard-Chebyshev method (Fukushima, 1997a). It obtains a global solution of ODE in the form of Chebyshev polynomial of large (> 1000) degree by applying the Picard iteration repeatedly. The iteration converges for smooth problems and/or perturbed dynamics. The method runs around 100-1000 times faster in the vector mode than in the scalar mode of a certain computer with vector processors (Fukushima, 1997b). The second is a parallelization of a symplectic integrator (Saha et al., 1997). It regards the implicit midpoint rules covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method is applicable to Hamiltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with more than 1000 PUs. The last is a parallelization of the extrapolation method (Ito and Fukushima, 1997). It performs trial integrations in parallel. Also the trial integrations are further accelerated by balancing computational load among PUs by the technique of folding. The method is all-purpose and achieves an acceleration factor of around 3.5 by using several PUs. Finally, we give a perspective on the parallelization of some implicit integrators which require multiple corrections in solving implicit formulas like the implicit Hermitian integrators (Makino and Aarseth, 1992), (Hut et al., 1995) or the implicit symmetric multistep methods (Fukushima, 1998), (Fukushima, 1999).

Timestep Acceleration of Waveform Relaxation

Dynamic iteration methods for treating certain classes of linear systems of differential equations are considered. It is shown that the discretized Picard--Lindelöf (waveform relaxation) iteration can be accelerated by solving the defect equations with a larger timestep, or by using a recursive procedure based on a succession of increasing timesteps. A discussion of convergence is presented, including analysis of a discrete smoothing property maintained by symmetric multistep methods applied to linear wave equations. Numerical experiments indicate that the method can speed convergence.

Stability of the symmetric multistep methods for the time-dependent Schrödinger equation

We study the stability of the explicit symmetric multistep scheme (ESM) as a numerical method of solving the time-dependent Schrödinger equation. We use the most general sets of the coefficients of ESM in order to find a set of coefficients that gives the widest range of stability.

Symmetric multistep methods for the numerical integration of planetary orbits

view Abstract Citations (124) References (14) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Symmetric Multistep Methods for the Numerical Integration of Planetary Orbits Quinlan, Gerald D. ; Tremaine, Scott Abstract The limit to the numerical accuracy of integrations of planetary orbits is set by the accumulation of round-off and truncation error. For the usual Stoermer multistep methods, even if steps are taken to reduce round-off error, truncation error still results in an energy error that grows linearly with time, which leads to a longitude error that grows quadratically with time. Here, 'symmetric' multistep methods are developed for which truncation leads to a longitude error that grows only linearly with time. The superiority of the symmetric methods over the Stoermer methods is illustrated by numerical examples. The optimum choice of the order and the coefficients of a symmetric multistep method for planetary integrations are discussed. Publication: The Astronomical Journal Pub Date: November 1990 DOI: 10.1086/115629 Bibcode: 1990AJ....100.1694Q Keywords: Computational Astrophysics; Numerical Integration; Orbit Calculation; Solar Orbits; Dynamic Stability; Iterative Solution; Linear Equations; Orbit Perturbation; Truncation Errors; Astronomy; SOLAR SYSTEM: GENERAL full text sources ADS |

The Exponential Accuracy of Fourier and Chebyshev Differencing Methods

It is shown that when differencing analytic functions using the pseudospectral Fourier or Chebyshev methods, the error committed decays to zero at an exponential rate.

A Sixth-order P-stable Symmetric Multistep Method for Periodic Initial-value Problems of Second-order Differential Equations

A Sixth-order <i>P</i>-stable Symmetric Multistep Method for Periodic Initial-value Problems of Second-order Differential Equations

Numerical integrators for stiff and highly oscillatory differential equations

Some L-stable fourth-order explicit one-step numerical integration formulas which require no matrix inversion are proposed to cope effectively with systems of ordinary differential equations with large Lipschitz constants (including those having highly oscillatory solutions). The implicit integration procedure proposed in Fatunla [11] is further developed to handle a larger class of stiff systems as well as those with highly oscillatory solutions. The same pair of nonlinear equations as in [11] is solved for the stiffness/oscillatory parameters. However, the nonlinear systems are transformed into linear forms and an efficient computational procedure is developed to obtain these parameters. The new schemes compare favorably with the backward differentiation formula (DIFSUB) of Gear [13], [14] and the blended linear multistep methods of Skeel and Kong [24], and the symmetric multistep methods of Lambert and Watson [17].