Partitioned Runge-Kutta Methods Research Articles

Discussion on: “An Optimal Control Approach for Hybrid Systems”

A finite volume approximation of the scalar hyperbolic conservation law or advection–diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge–Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge–Kutta methods are used. An adaptive Runge–Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge–Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem.

Global modified Hamiltonian for constrained symplectic integrators

We prove that the numerical solution of partitioned Runge-Kutta methods applied to constrained Hamiltonian systems (e.g., the Rattle algorithm or the Lobatto IIIA–IIIB pair) is formally equal to the exact solution of a constrained Hamiltonian system with a globally defined modified Hamiltonian. This property is essential for a better understanding of their longtime behaviour. As an illustration, the equations of motion of an unsymmetric top are solved using a parameterization with Euler parameters.

Multirate Partitioned Runge-Kutta Methods

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.

Order conditions for partitioned Runge-Kutta methods

We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.

Error of Partitioned Runge-Kutta Methods for Multiple Stiff Singular Perturbation Problems

The main purpose of this paper is to deal with error behaviour of partitioned Runge-Kutta methods for one-parameter multiple stiff singular perturbation problems whose stiffness is caused by a small parameter e and some other factors and to present some quantitative convergence results.

Formal series and numerical integrators, Part II: Application to index-2 differential-algebraic systems

This paper is in two parts. In the first part, we presented a unified approach for deriving and dealing with the formal series that arise when studying the convergence of numerical integrators for systems of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). This general approach was applied for systems of ODEs. The particular case of Hamiltonian systems was studied in some detail. In this the second part, we study in a unified way convergence of a general class of Runge-Kutta type methods (the partitioned Runge-Kutta methods) for systems of index 2 DAEs in Hessenberg form, using tools developed in the first part to handle the relevant formal expansions, the DA2-series. Our approach allows us to obtain sharp estimates for the global errors, even in cases where the method does not satisfy the algebraic constraints, the errors for the algebraic variables affect the approximations to the differential variables, and inconsistent initial values are used.

Runge-kutta methods for hamiltonian systems in non-standard symplectic two-form

Runge-Kutta methods are applied to Hamiltonian systems on Poisson manifolds with a nonstandard symplectic two-form. It has been shown that the Gauss Legendre Runge-Kutta (GLRK) methods and combination of the partitioned Runge-Kutta methods of Lobatto IIIA and IIIb type are symplectic up to the second order in terms of the step size. Numerical results on Lotka-Volterra and Kermack-McKendrick epidemic disease model reveals that the application of the symplectic Runge-Kutta methods preserves the integral invariants of the underlying system for long-time computations.

Numerical solution of viscoplastic constitutive equations with internal state variables Part I: algorithms and implementation

This article treats the initial-boundary-value problem of viscoplasticity using unified constitutive models without a yield surface. Semi-discretization with the finite element method (FEM) leads to a system of differential-algebraic equations (DAE) with strongly non-linear evolution equations for the internal state variables. A special family of partitioned Runge-Kutta methods is introduced which allows an efficient time integration of the semidiscrete system. Coefficients for methods of order one, two, and three are given. Finally, numerical results for some two- and three-dimensional examples using the model of Hart are presented. In a second part we will give the theoretical background and state a proof of convergence for the algorithm presented in this paper.

On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems

In this paper we generalize the class of explicit partitioned Runge-Kutta (PRK) methods for separable Hamiltonian systems to systems with holonomic constraints. For a convenient analysis of such schemes, we first generalize the backward error analysis for systems in ${\Bbb R}^m$ to systems on manifolds embedded in ${\Bbb R}^m$ . By applying this analysis to constrained PRK methods, we prove that such methods will, in general, suffer from order reduction as well-known for higher-index differential-algebraic equations. However, this order reduction can be avoided by a proper modification of the standard PRK methods. This modification increases the number of projection steps onto the constraint manifold but leaves the number of force evaluations constant. We also give a numerical comparison of several second, fourth, and sixth order methods.

Partitioned Runge-Kutta methods as phase volume preserving integrators

We analyse the property of phase volume preservation for the partitioned Runge-Kutta methods. A family of explicit methods known to be symplectic is demonstrated to possess the volume preservation property as well. Runge-Kutta-Nyström methods are also considered.

Symplectic Partitioned Runge–Kutta Methods for Constrained Hamiltonian Systems

This article deals with the numerical treatment of Hamiltonian systems with holonomic constraints. A class of partitioned Runge–Kutta methods, consisting of the couples of s-stage Lobatto IIIA and Lobatto IIIB methods, has been discovered to solve these problems efficiently. These methods are symplectic, preserve all un-derlying constraints, and are superconvergent with order $2s - 2$. For separable Hamiltonians of the form $H(q,p) = \frac{1}{2}p^T M^{ - 1} p + U(q)$ the Rattle algorithm based on the Verlet method was up to now the only known symplectic method preserving the constraints. In fact this method turns out to be equivalent to the 2-stage Lobatto IIIA–IIIB method of order 2. Numerical examples have been performed which illustrate the theoretical results.

Momentum conserving symplectic integrators

In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL( n).

Partitioned Runge-Kutta Methods for Separable Hamiltonian Problems

Partitioned Runge-Kutta Methods for Separable Hamiltonian Problems

Partitioned Runge-Kutta methods for separable Hamiltonian problems

Separable Hamiltonian systems of differential equations have the form d p / d t = − ∂ H / ∂ q d{\mathbf {p}}/dt = - \partial H/\partial {\mathbf {q}} , d q / d t = ∂ H / ∂ p d{\mathbf {q}}/dt = \partial H/\partial {\mathbf {p}} , with a Hamiltonian function H that satisfies H = T ( p ) + V ( q ) H = T({\mathbf {p}}) + V({\mathbf {q}}) (T and V are respectively the kinetic and potential energies). We study the integration of these systems by means of partitioned Runge-Kutta methods, i.e., by means of methods where different Runge-Kutta tableaux are used for the p and q equations. We derive a sufficient and "almost" necessary condition for a partitioned Runge-Kutta method to be canonical, i.e., to conserve the symplectic structure of phase space, thereby reproducing the qualitative properties of the Hamiltonian dynamics. We show that the requirement of canonicity operates as a simplifying assumption for the study of the order conditions of the method.

Explicit canonical methods for Hamiltonian systems

We consider canonical partitioned Runge-Kutta methods for separable Hamiltonians H = T ( p ) + V ( q ) H = T(p) + V(q) and canonical Runge-Kutta-Nyström methods for Hamiltonians of the form H = 1 2 p T M − 1 p + V ( q ) H = \frac {1}{2}{p^{\text {T}}}{M^{ - 1}}p + V(q) with M a diagonal matrix. We show that for explicit methods there is great simplification in their structure. Canonical methods of orders one through four are constructed. Numerical experiments indicate the suitability of canonical numerical schemes for long-time integrations.

Partitioned Runge-Kutta methods with stiffness detection and stepsize control

For the numerical solution of autonomous initial value problems partitioned Runge-Kutta methods are studied, consisting of anA-stable Rosenbrock-Wanner method for the treatment of the stiff components and a customary Runge-Kutta method for the nonstiff part. The equations of condition including the coupling conditions are presented. An automatic stiffness detection and a stepsize control for two different algorithms are developed. The first algorithm PRK4 of order (3)4 possesses an automatic componentwise stiffness detection. The second algorithm RKF4RW treats the whole system as nonstiff or as stiff. RKF4RW is based on the well-known Runge-Kutta-Fehlberg pair of order 4(5). Additionally a ROW method of order (3)4 with anA-stable fourth order approximation is embedded. Usually RKF4RW is the faster algorithm and the system stiffness detection works more reliable than the componentwise stiffness detection in PRK4.