Energy-preserving continuous-stage partitioned Runge-Kutta methods

We consider the application of partitioned Runge-Kutta (PRK) methods to non-autonomous Hamiltonian systems. Necessary and sufficient conditions for the symplecticness of PRK methods are given, more particularly for two low order PRK methods: the partitioned (explicit-implicit) Euler method and the 2-stage Lobatto IIIA-B PRK method. Both methods are often the basis of composition schemes of higher order. In particular for irreducible PRK methods we show the necessity for the nodes of the two underlying PRK methods to be equal.

The coupling of subsystems in a hierarchical modelling approach leads to different time constants in the dynamical simulation of technical systems. Multirate schemes exploit the different time scales by using different time steps for the subsystems. The stiffness of the system or at least of some subsystems in chemical reaction kinetics or network analysis, for example, forbids the use of explicit integration schemes. To cope with stiff problems, we introduce multirate schemes based on partitioned Runge—Kutta methods which avoid the coupling between active and latent components based on interpolating and extrapolating state variables. Order conditions and test results for such a lower order MPRK method are presented.

Previously, it has been shown that discretizing a multi-Hamiltonian PDE in space and time with partitioned Runge–Kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law. However, these previous studies use the same partitioning of the variables into two parts in both space and time. This gives rise to a large number of cases to be considered, each with its own set of conditions to be satisfied. We present here a much simpler set of conditions, covering all of these cases, where the variables are partitioned independently in space and time into an arbitrary number of parts. In general, it is not known when such a discretization of a multi-Hamiltonian PDE will give rise to a well-defined numerical integrator. However, a numerical integrator that is explicit will typically be well defined. In this paper, we give sufficient conditions on a multi-Hamiltonian PDE for a Lobatto IIIA–IIIB discretization in space to give rise to explicit ODEs and an algorithm for constructing these ODEs.

Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge–Kutta discretizations is well-defined if and only if the partitioned Runge-Kutta methods are symplectic in space and time.

A method for solving bang-bang and singular optimal control problems using the integral form of Legendre-Gauss-Lobatto collocation is developed. The optimal control problem is developed in multiple-phase form in order to accurately capture discontinuities in the control. This multiple-phase optimal control problem is approximated using a recently developed integral form of Legendre-Gauss-Lobatto (LGL-I) collocation method, where the LGL-I method has the key feature that it approximates the state and control at both endpoints of the interval. In addition, recent research has shown that this new LGL-I method is a partitioned Runge-Kutta method that exhibits superconvergence at the mesh points. The method developed in this paper is applied to a minimum-time axisymmetric reorientation problem with bang-bang and singular control structure, where it is found that the LGL-I method produces accurate solutions.

High-order stochastic symplectic partitioned Runge-Kutta methods for stochastic Hamiltonian systems with additive noise

Trigonometrically fitted symplectic Partitioned Runge Kutta (EFSPRK) methods for the numerical integration of Hamoltonian systems with oscillatory solutions are derived. These methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions sin(wx),cos(wx), w∈R. We modify a fifth order symplectic PRK method with six stages so to derive an exponentially fitted SPRK method. The methods are tested on the numerical integration of the two body problem.

This paper focuses on the formulation of variational integrators (VIs) for finite‐dimensional thermo‐elastic systems without heat conduction. The dynamics of these systems happen to have a Hamiltonian structure, after thermal displacements are introduced. It is then possible to formulate integrators by taking advantage of standard methods in VI. The class of integrators we construct have some remarkable features: (a) they are symplectic, (b) they exactly conserve the entropy of the system, or in other words, they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes, (c) they nearly exactly conserve the value of the energy for very long times and (d) they exactly conserve linear and angular momentum. We first describe how to adapt any VI for the classical mechanical systems to integrate adiabatic thermo‐elastic ones, and then formulate three new types of integrators. The first class, based on a generalized trapezoidal rule, gives rise to two first order, explicit integrators, and a second order, implicit one. By composing then the two first‐order integrators we construct a second order, explicit one. Finally, we formulate a fourth order, implicit integrator, which is a symplectic partitioned Runge–Kutta method. The performance of these new algorithms is showcased through numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.

This work considers multirate generalized-structure additively partitioned Runge–Kutta methods for solving stiff systems of ordinary differential equations with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

A mathematically rigorous methodology for embedding the governing equations of molecular dynamics in the formalism of the finite element method is presented. Only one generalized finite element type is needed to cover all different types of existing interatomic potentials. The finite element type is simply specified by two parameters characterizing the type of the interatomic potential to be considered. Built on this formulation a partitioned Runge–Kutta method—summarizing a wide range of explicit and implicit, single- and multi-stage, lower and higher order time integration schemes—is embedded in a unified manner. The required finite element residual vector and the related Jacobian matrix are stated explicitly. The related FE-mesh coincides with the neighborhood lists used in standard molecular dynamics enabling the use of common tools. The range, versatility and performance of the proposed finite element formulation have been demonstrated by means of several numerical examples.

In this paper, we construct second order symplectic and multisymplectic integrators for the “good” Boussineq equation using the two-stage Lobatto IIIA-IIIB partitioned Runge–Kutta method, which yield an explicit scheme and is equivalent to the classical central difference approximation to the second order spatial derivative. Numerical dispersion properties and the stability of both integrators are investigated. Numerical results for different solitary wave solutions confirm the excellent long time behavior of symplectic and multisymplectic integrators by preserving local and global energy and momentum.

In this paper we present second, third and fourth order symplectic partitioned Runge-Kutta (PRK) methods with several stages. We apply all methods to the numerical integration of the harmonic oscillator, the pendulum and Kepler problems.

Multirate schemes for conservation laws or convection-dominated problems seem to come in two flavors: schemes that are locally inconsistent, and schemes that lack mass-conservation. In this paper these two defects are discussed for one-dimensional conservation laws. Particular attention will be given to monotonicity properties of the multirate schemes, such as maximum principles and the total variation diminishing (TVD) property. The study of these properties will be done within the framework of partitioned Runge-Kutta methods. It will also be seen that the incompatibility of consistency and mass-conservation holds for ‘genuine’ multirate schemes, but not for general partitioned methods.KeywordsMonotonicity PropertyTotal Variation DiminishMonotonicity AssumptionStrong Stability PreserveHigh Resolution SchemeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Multirate GARK schemes define a multirate extension of GARK schemes, generalized additive Runge-Kutta schemes. In contrast to additive schemes, GARK schemes allow for different stage values as arguments of different components of the right hand side. They introduce additional flexibility when compared to traditional partitioned Runge-Kutta methods, and therefore offer additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.

A framework is presented allowing dual-rate numerical integration of the equations of mechanical system dynamics to be considered as a form of Partitioned Runge-Kutta (PRK) integration. Certain coefficients of a PRK integrator are set to zero, so that Runge-Kutta integrators that constitute the PRK integrator can be made to have different numbers of stages. As a result, one Runge-Kutta integrator requires fewer function evaluations than the other does, which is a form of dual-rate integration. Well-established order of accuracy theory for PRK integrators is used to develop a rigorous methodology for designing explicit PRK dual-rate integrators. Stabilized Runge-Kutta theory developed for single-rate Runge-Kutta integrators is combined with PRK integrator theory to design PRK dual-rate integrators with the largest possible stability regions. Dual-rate PRK integrators created using these approaches are used to simulate the dynamics of vehicle systems that contain subsystems with higher frequency response characteristics than do the basic vehicle subsystems. #The work presented here was published in 2000 as Siddhartha S. Shome's Ph.D. dissertation at the University of Iowa, Dual-Rate Integration Using Partitioned Runge-Kutta Methods for Mechanical Systems with Interacting Subsystems. Keywords: Partitioned Runge-Kutta methodsOrdinary differential equations Acknowledgments Grateful acknowledgments are due to Dr. Dario Solis of the National Advanced Driving Simulator and Simulation Center at the University of Iowa for many useful discussions. This material is partly based upon work supported by the National Science Foundation under Grant No. 9983708. Notes #The work presented here was published in 2000 as Siddhartha S. Shome's Ph.D. dissertation at the University of Iowa, Dual-Rate Integration Using Partitioned Runge-Kutta Methods for Mechanical Systems with Interacting Subsystems.