Ander Murua

Sergio Blanes, Fernando Casas, Ariadna Farres et al.

We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very accurate results over a large time span. We derive in a systematic way an independent set of necessary and sufficient conditions to be satisfied by the coefficients of splitting methods to achieve a prescribed order of accuracy. Splitting methods satisfying such (generalized) order conditions are appropriate in particular for the numerical simulation of the Solar System described in Jacobi coordinates. We show that, when using Poincaré Heliocentric coordinates, the same order of accuracy may be obtained by imposing an additional polynomial equation on the coefficients of the splitting method. We construct several splitting methods appropriate for each of the two sets of coordinates by solving the corresponding systems of polynomial equations and finding the optimal solutions. The experiments reported here indicate that the efficiency of our new schemes is clearly superior to previous integrators when high accuracy is required.

Ariadna Farrés, Jacques Laskar, Sergio Blanes et al.

Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term studies in the Solar System. These comparisons are made in Jacobi and Heliocentric coordinates and the implementation of the algorithms is fully detailed for practical use. We conclude that high order integrators should be privileged, with a preference for the new $(10,6,4)$ method of (Blanes et al., 2012)

Sergio Blanes, Fernando Casas, Ander Murua

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.

Ander Murua, J. M. Sanz-Serna

We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves by means of a simple rule. While word series have appeared before in the literature, extended word series are introduced in this paper. We exemplify the use of extended word series by studying the reduction to normal form and averaging of some perturbed integrable problems. We also provide a detailed analysis of the behaviour of splitting numerical methods for those problems.

Sergio Blanes, Fernando Casas, Philippe Chartier et al.

We are concerned with the numerical solution obtained by splitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this second-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, methods of orders 3 to 14 by using the Suzuki--Yoshida triple (and quadruple) jump composition procedure have been explicitly built. Here we reconsider this technique and show that it is inherently bounded to order 14 and clearly sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct methods of orders 6 and 8 that are the most accurate ones available at present time, even when low accuracies are desired. We also show that, in the general case, 14 is not an order barrier for splitting methods with complex coefficients with positive real part by building explicitly a method of order 16 as a composition of methods of order 8.

Ander Murua, J. M. Sanz-Serna

We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the tech- nique applied here, the tasks of constructing the averaged system and the associ- ated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of sim- ple recursions. As distinct from the situation with other approaches, with word- series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approxima- tions to the true dynamics.

Mikel Antoñana, Joseba Makazaga, Ander Murua

We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.

Sergio Blanes, Fernando Casas, Ander Murua

A typical procedure to integrate numerically the time dependent Schrö\-din\-ger equation involves two stages. In the first one carries out a space discretization of the continuous problem. This results in the linear system of differential equations $i du/dt = H u$, where $H$ is a real symmetric matrix, whose solution with initial value $u(0) = u_0 \in \mathbb{C}^N$ is given by $u(t) = \e^{-i t H} u_0$. Usually, this exponential matrix is expensive to evaluate, so that time stepping methods to construct approximations to $u$ from time $t_n$ to $t_{n+1}$ are considered in the second phase of the procedure. Among them, schemes involving multiplications of the matrix $H$ with vectors, such as Lanczos and Chebyshev methods, are particularly efficient. In this work we consider a particular class of splitting methods which also involves only products $Hu$. We carry out an error analysis of these integrators and propose a strategy which allows us to construct different splitting symplectic methods of different order (even of order zero) possessing a large stability interval that can be adapted to different space regularity conditions and different accuracy ranges of the spatial discretization. The validity of the procedure and the performance of the resulting schemes are illustrated on several numerical examples.

Robert I McLachlan, Ander Murua

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the `Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra $\mathcal X$, spanned by `modified' potential energies isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with $\mathcal X$. We calculate the dimensions $c_n$ of its homogeneous subspaces and determine the value of its entropy $\lim_{n\to\infty} c_n^{1/n}$. It is $1.8249\dots$, a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.

Mikel Antoñana, Joseba Makazaga, Ander Murua

We propose a family of integrators, Flow-Composed Implicit Runge-Kutta (FCIRK) methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step of an implicit Runge-Kutta (IRK) method applied to a transformed system. The resulting integration schemes are symplectic when both the perturbation and the unperturbed part are Hamiltonian and the underlying IRK scheme is symplectic. In addition, they are symmetric in time (resp. have order of accuracy $r$) if the underlying IRK scheme is time-symmetric (resp. of order $r$). The proposed new methods admit mixed precision implementation that allows us to efficiently reduce the effect of round-off errors. We particularly focus on the potential application to long-term solar system simulations, with the equations of motion of the solar system rewritten as a Hamiltonian perturbation of a system of uncoupled Keplerian equations. We present some preliminary numerical experiments with a simple point mass Newtonian 10-body model of the solar system (with the sun, the eight planets, and Pluto) written in canonical heliocentric coordinates.

Mikel Antoñana, Joseba Makazaga, Ander Murua

We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular attention to symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For a $s$-stage IRK scheme used to integrate a $d$-dimensional system of ordinary differential equations, the application of simplified versions of Newton iterations requires solving at each step several linear systems (one per iteration) with the same $sd \times sd$ real coefficient matrix. We propose rewriting such $sd$-dimensional linear systems as an equivalent $(s+1)d$-dimensional systems that can be solved by performing the LU decompositions of $[s/2] +1$ real matrices of size $d \times d$. We present a C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the linear system and that takes special care in reducing the effect of round-off errors. We report some numerical experiments that demonstrate the reduced round-off error propagation of our implementation.

Mikel Antoñana, Joseba Makazaga, Ander Murua

The main purpose of this work is to present a SIMD-vectorized implementation of the symplectic 16th-order 8-stage implicit Runge-Kutta integrator based on collocation with Gauss-Legendre nodes (IRKGL16-SIMD), and to show that it can outperform state-of-the-art symplectic explicit integrators for high-precision numerical integrations (in double-precision floating-point arithmetic) of non-stiff Hamiltonian ODE systems. Our IRKGL16-SIMD integrator leverages Single Instruction Multiple Data (SIMD) based parallelism (in a way that is transparent to the user) to significantly enhance the performance of the sequential IRKGL16 implementation. We present numerical experiments comparing IRKGL16-SIMD with state-of-the-art high-order explicit symplectic methods for the numerical integration of several Hamiltonian systems in double-precision floating-point arithmetic.

Fernando Casas, Ander Murua

Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes are available that achieve high accuracy while preserving key qualitative properties of the underlying dynamical system, and are successfully used across a broad range of fields. In this work, we present a systematic analysis of both local and global errors arising from arbitrary splitting methods applied to unitary problems. Two complementary types of error estimates are derived. The first is expressed in terms of operator norms, while the second is formulated using norms of commutators and can, under suitable assumptions, be extended to certain classes of unbounded operators. Special attention is devoted to the case where only two operators are involved. The theoretical results are illustrated by deriving explicit error bounds for some representative schemes.

Jesus Maria Sanz-Serna, Ander Murua

This paper provides a brief history of B-series and the associated Butcher group and presents the new theory of word series and extended word series. B-series (Hairer and Wanner 1976) are formal series of functions parameterized by rooted trees. They greatly simplify the study of Runge-Kutta schemes and other numerical integrators. We examine the problems that led to the introduction of B-series and survey a number of more recent developments, including applications outside numerical mathematics. Word series (series of functions parameterized by words from an alphabet) provide in some cases a very convenient alternative to B-series. Associated with word series is a group G of coefficients with a composition rule simpler than the corresponding rule in the Butcher group. From a more mathematical point of view, integrators, like Runge-Kutta schemes, that are affine equivariant are represented by elements of the Butcher group, integrators that are equivariant with respect to arbitrary changes of variables are represented by elements of the word group G.

Sergio Blanes, Fernando Casas, Ander Murua

Splitting methods for the numerical integration of differential equations of order greater than two involve necessarily negative coefficients. This order barrier can be overcome by considering complex coefficients with positive real part. In this work we review the composition technique used to construct methods of this class, propose new sixth-order integrators and analyze their main features on a pair of numerical examples, in particular how the errors are propagated along the evolution.