Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

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

Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

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 × 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 × 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.Project of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU). Project MTM2013-46553-C3-2-P from Spanish Ministry of Economy and Trade. Consolidated Research Group IT649-13 from the Basque Government

Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes

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.Ministerio de Economía y Comercio: proyecto MTM2013-46553-C3-2-P, Spanish Ministry of Economy and Competitiveness: project MTM2016-76329-R “IMAGEARTH”, Basque Government: Consolidated Research Group IT649-1

Efficient implementation of geometric integrators for separable Hamiltonian problems

We here investigate the efficient implementation of the energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) recently introduced for the numerical solution of Hamiltonian problems. In this note, we describe an iterative procedure, based on a triangular splitting, for solving the generated discrete problems, when the problem at hand is separable.Comment: 4 page

Verification of Planetary Protection Requirements with Symplectic Methods and Monte Carlo Line Sampling

Verification of the compliance to planetary protection requirements is an important task of interplanetary mission design, aiming to reduce the risk of biological contamination of scientifically interesting celestial bodies. This kind of analysis requires efficient and reliable numerical tools to propagate uncertainties over times up to 100 years with high precision. This paper presents a plan to improve the techniques used for planetary protection analysis in the SNAPPshot numerical tool developed at the University of Southampton for an ESA study. The Line Sampling method is presented as an alternative Monte Carlo approach to sample more efficiently the initial uncertainties, reducing the computational effort to estimate the probability of impact between uncontrolled objects and a celestial body. Symplectic integration methods are introduced as a strategy to obtain a more accurate propagation of the spacecraft trajectory starting from the initial conditions, thanks to their formulation that includes the conservation of total energy. Preliminary results are included to show the advantages and the current limitations of the proposed approaches

Almost Symplectic Runge-Kutta Schemes for Hamiltonian Systems

Symplectic Runge-Kutta schemes for the integration of general Hamiltonian systems are implicit. In practice one has to solve the implicit algebraic equations using some iterative approximation method, in which case the resulting integration scheme is no longer symplectic. In this paper we first analyze the preservation of the symplectic structure under two popular approximation schemes, fixed-point iteration and Newton's method, respectively. Error bounds for the symplectic structure are established when N fixed-point iterations or N iterations of Newton's method are used. The implications of these results for the implementation of symplectic methods are discussed and then explored through extensive numerical examples. Numerical comparisons with non-symplectic Runge-Kutta methods and pseudo-symplectic methods are also presented