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 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

An implicit symplectic solver for high-precision long term integrations of the Solar System

Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of round-off errors. In addition, unlike typical explicit symplectic integrators for near Keplerian problems, FCIRK16 is able to integrate problems with arbitrary perturbations (non necessarily split as a sum of integrable parts). We present a novel analysis of the effect of close encounters in the leading term of the local discretization errors of our integrator. Based on that analysis, a mechanism to detect and refine integration steps that involve close encounters is incorporated in our code. That mechanism allows FCIRK16 to accurately resolve close encounters of arbitrary bodies. We illustrate our treatment of close encounters with the application of FCIRK16 to a point mass Newtonian 15-body model of the Solar System (with the Sun, the eight planets, Pluto, and five main asteroids) and a 16-body model treating the Moon as a separate body. We also present some numerical comparisons of FCIRK16 with a state-of-the-art high order explicit symplectic scheme for 16-body model that demonstrate the superiority of our integrator when very high precision is required

An implicit symplectic solver for high-precision long term integrations of the Solar System

We present FCIRK16, a 16th-order implicit symplectic integrator for long-term high precision Solar System simulations. Our integrator takes advantage of the near-Keplerian motion of the planets around the Sun by alternating Keplerian motions with corrections accounting for the planetary interactions. Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of round-off errors. In addition, unlike typical explicit symplectic integrators for near Keplerian problems, FCIRK16 is able to integrate problems with arbitrary perturbations (non necessarily split as a sum of integrable parts). We present a novel analysis of the effect of close encounters in the leading term of the local discretization errors of our integrator. Based on that analysis, a mechanism to detect and refine integration steps that involve close encounters is incorporated in our code. That mechanism allows FCIRK16 to accurately resolve close encounters of arbitrary bodies. We illustrate our treatment of close encounters with the application of FCIRK16 to a point mass Newtonian 15-body model of the Solar System (with the Sun, the eight planets, Pluto, and five main asteroids) and a 16-body model treating the Moon as a separate body. We also present some numerical comparisons of FCIRK16 with a state-of-the-art high order explicit symplectic scheme for 16-body model that demonstrate the superiority of our integrator when very high precision is required.Consolidated Research Group MATHMODE (IT1294-19

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances