Partitioned Runge Kutta Research Articles

Symmetric partitioned Runge–Kutta methods for differential equations on Lie groups

In this paper, we develop a higher order symmetric partitioned Runge–Kutta method for a coupled system of differential equations on Lie groups. We start with a discussion on partitioned Runge–Kutta methods on Lie groups of arbitrary order. As symmetry is not met for higher orders, we generalize the method to a symmetric partitioned Runge–Kutta (SPRK) scheme. Furthermore, we derive a set of coefficients for convergence order 4. The SPRK integration method can be used, for example, in simulations of quantum field theories. Finally, we compare the new SPRK scheme numerically with the Störmer–Verlet scheme, one of the state-of-the-art schemes used in this subject.

Phase fitted symplectic partitioned Runge–Kutta methods for the numerical integration of the Schrödinger equation

In this work we consider explicit symplectic partitioned Runge–Kutta methods with five stages for problems with separable Hamiltonian. We construct three new methods, one with constant coefficients of eight phase-lag order and two phase-fitted methods.

Symplectic partitioned Runge–Kutta methods with the phase-lag property

In this work specially tuned symplectic partitioned Runge–Kutta (SPRK) methods with minimum phase-lag and phase fitted have been considered. The general framework for constructing SPRK methods with minimum phase-lag is given.

Symplectic partitioned Runge–Kutta methods for two-dimensional numerical model of ground penetrating radar

Simulation of ground penetrating radar (GPR) wave propagation in two-dimensional (2-D) subsurface structure is developed using the symplectic partitioned Runge–Kutta (SPRK) method. A transmitting boundary is implemented to absorb waves at the edges of the modeling. For the 2-D case, the SPRK schemes require only two functions for the complete description of the electromagnetic field. To verify the performance of the proposed algorithms, results from comparisons between the SPRK schemes and the standard FDTD method are presented. In addition, a complicated subsurface structure model is considered. The wiggle trace profile of this model is obtained from forward simulation using a 2-order SPRK method.

TWO NEW PHASE-FITTED SYMPLECTIC PARTITIONED RUNGE–KUTTA METHODS

New symplectic Partitioned Runge–Kutta (SPRK) methods with phase-lag of order infinity are derived in this paper. Specifically two new symplectic methods are constructed with second and third algebraic order. The methods are tested on the numerical integration of Hamiltonian problems and on the estimation of the eigenvalues of the Schrödinger equation.

A nearly analytic symplectically partitioned Runge-Kutta method for 2-D seismic wave equations

SUMMARY In this paper, we develop a new nearly analytic symplectically partitioned Runge–Kutta (NSPRK) method for numerically solving elastic wave equations. In this method, we first transform the elastic wave equations into a Hamiltonian system, and use the nearly analytic discrete operator to approximate the high-order spatial differential operators, and then we employ the partitioned second-order symplectic Runge–Kutta method to numerically solve the resulted semi-discrete Hamiltonian ordinary differential equations (ODEs). We investigate in great detail on the properties of the NSPRK method that includes the stability condition for the P–SV wave in a 2-D homogeneous isotropic medium, the computational efficiency, and the numerical dispersion relation for the 2-D acoustic case. Meanwhile, we apply the NSPRK to simulate the elastic wave propagating in several multilayer models with both strong velocity contrasts and fluctuating interfaces. Both theoretical analysis and numerical results show that the NSPRK can effectively suppress the numerical dispersion resulted from the discretization of the wave equations, and more importantly, it preserves the symplecticity structure for long-time computation. In addition, numerical experiments demonstrate that the NSPRK is effective to combine the split perfectly matched layer boundary conditions to take care of the reflections from the artificial boundaries.

Variational time integrators for finite‐dimensional thermo‐elasto‐dynamics without heat conduction

AbstractThis 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.

Explicit symplectic partitioned Runge–Kutta–Nyström methods for non-autonomous dynamics

We consider explicit symplectic partitioned Runge–Kutta (ESPRK) methods for the numerical integration of non-autonomous dynamical systems. It is known that, in general, the accuracy of a numerical method can diminish considerably whenever an explicit time dependence enters the differential equations and the order reduction can depend on the way the time is treated. In the present paper, we demonstrate that explicit symplectic partitioned Runge–Kutta–Nyström (ESPRKN) methods specifically designed for second order differential equations d d t ( M − 1 q ˙ ) = f ( q , t ) , undergo an order reduction when M = M ( t ) , independently of the way the time is approximated. Furthermore, by means of symmetric quadrature formulae of appropriate order, we propose a different but still equivalent formulation of the original non-autonomous problem that treats the time as two added coordinates of an enlarged differential system. In so doing, the order reduction is avoided as confirmed by the presented numerical tests.

Preservation of stability properties near fixed points of linear Hamiltonian systems by symplectic integrators

Based on reasonable testing model problems, we study the preservation by symplectic Runge–Kutta method (SRK) and symplectic partitioned Runge–Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The structure-preservation region provides a practical criterion for choosing step-size in symplectic computation. Examples are given to justify the investigation.

Linear Stability of Partitioned Runge–Kutta Methods

We study the linear stability of partitioned Runge–Kutta (PRK) methods applied to linear separable Hamiltonian ODEs and to the semidiscretization of certain Hamiltonian PDEs. We extend the work of Jay and Petzold [Highly Oscillatory Systems and Periodic Stability, Preprint 95-015, Army High Performance Computing Research Center, Stanford, CA, 1995] by presenting simplified expressions of the trace of the stability matrix, ${tr}M_s$, for the Lobatto IIIA–IIIB family of symplectic PRK methods. By making the connection to Padé approximants and continued fractions, we study the asymptotic behavior of ${tr}M_s(\omega)$ as a function of the frequency $\omega$ and stage number s.

Partitioned Runge–Kutta–Chebyshev Methods for Diffusion-Advection-Reaction Problems

An integration method based on Runge–Kutta–Chebyshev (RKC) methods is discussed which has been designed to treat moderately stiff and nonstiff terms separately. The method, called partitioned Runge–Kutta–Chebyshev (PRKC), is a one-step, partitioned RK method of second order. It belongs to the class of stabilized methods, namely explicit RK methods possessing extended real stability intervals. The aim of the PRKC method is to reduce the number of function evaluations of the nonstiff terms and to get a nonzero imaginary stability boundary.

Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control

Abstract We are concerned with the discretization of optimal control problems when a Runge–Kutta scheme is selected for the related Hamiltonian system. It is known that Lagrangian’s first order conditions on the discrete model, require a symplectic partitioned Runge–Kutta scheme for state–costate equations. In the present paper this result is extended to growth models, widely used in Economics studies, where the system is described by a current Hamiltonian. We prove that a correct numerical treatment of the state–current costate system needs Lawson exponential schemes for the costate approximation. In the numerical tests a shooting strategy is employed in order to verify the accuracy, up to the fourth order, of the innovative procedure we propose.

Lobatto IIIA–IIIB discretization of the strongly coupled nonlinear Schrödinger equation

In this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrödinger equation based on the two-stage Lobatto IIIA–IIIB partitioned Runge–Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass.

Semi-explicit symplectic partitioned Runge–Kutta Fourier pseudo-spectral scheme for Klein–Gordon–Schrödinger equations

In the manuscript, we propose an explicit symplectic partitioned Runge–Kutta Fourier pseudo-spectral (SPRK-FPS) scheme for the coupled Klein–Gordon–Schrödinger (KGS) equation. It is explicit in term of iteration since it is only required to solve two linear algebraic equations at each marching time step. Furthermore, it does not only symplectic geometric structure-preserving, but also charge-preserving and energy-preserving. The numerical results are in good agreement with the theoretical analysis.

Symplectic Partitioned Runge–Kutta methods with minimal phase-lag

Symplectic Partitioned Runge–Kutta (SPRK) methods with minimal phase-lag are derived. Specifically two new symplectic methods are constructed of second and third order with fifth phase-lag order. The methods are tested on the numerical integration of Hamiltonian problems and the Schrödinger equation.

Symplectic discretization for spectral element solution of Maxwell's equations

Applying the spectral element method (SEM) based on the Gauss–Lobatto–Legendre (GLL) polynomial to discretize Maxwell's equations, we obtain a Poisson system or a Poisson system with at most a perturbation. For the system, we prove that any symplectic partitioned Runge–Kutta (PRK) method preserves the Poisson structure and its implied symplectic structure. Numerical examples show the high accuracy of SEM and the benefit of conserving energy due to the use of symplectic methods.

Two new solutions to the third-order symplectic integration method

Two new solutions are obtained for the symplecticity conditions of explicit third-order partitioned Runge–Kutta time integration method. One of them has larger stability limit and better dispersion property than the Ruth's method.

A discrete action principle for electrodynamics and the construction of explicit symplectic integrators for linear, non-dispersive media

In this work, we derive a discrete action principle for electrodynamics that can be used to construct explicit symplectic integrators for Maxwell’s equations. Different integrators are constructed depending on the choice of discrete Lagrangian used to approximate the action. By combining discrete Lagrangians in an explicit symplectic partitioned Runge–Kutta method, an integrator capable of achieving any order of accuracy is obtained. Using the von Neumann stability analysis, we show that the integrators greatly increase the numerical stability and reduce the numerical dispersion compared to other methods. For practical purposes, we demonstrate how to implement the integrators using many features of the finite-difference time-domain method. However, our approach is also applicable to other spatial discretizations, such as those used in finite element methods. Using this implementation, numerical examples are presented that demonstrate the ability of the integrators to efficiently reduce and maintain a minimal amount of numerical dispersion, particularly when the time-step is less than the stability limit. The integrators are therefore advantageous for modeling large, inhomogeneous computational domains.

Numerical integration of relativistic equations of motion for Earth satellites

The equations of motion proposed by Brumberg for an artificial satellite around the Earth (Celest Mech Dyn Astron 88:209, 2004), in which the relativistic effects due to the Earth’s oblatness and the gravitational action caused by a third body are added to those perturbations considered in the International Earth Rotation and Reference System Service (2003) convention, are here integrated numerically. To compute the solution of the time-dependent Langrangian system for a gravitational satellite–Earth–Sun model we consider a six-order partitioned Runge–Kutta integrator, whose coefficients satisfy the condition of symplecticity. A comparison with the classical Adams–Basforth–Moulton method allows to verify the good-performance of the partitioned Runge–Kutta method both in the description of the evolution of the satellite energy and in the efficiency of the method when applied to a long-term integration.

Generating functions of multi-symplectic RK methods via DW Hamilton–Jacobi equations

The De Donder–Weyl (DW) Hamilton–Jacobi equation is investigated in this paper, and the connection between the DW Hamilton–Jacobi equation and multi-symplectic Hamiltonian system is established. Based on the DW Hamilton–Jacobi theory, generating functions for multi-symplectic Runge–Kutta (RK) methods and partitioned Runge–Kutta (PRK) methods are presented.