Symplectic Runge-Kutta Methods Research Articles

Diagonally implicit trigonometrically fitted symplectic Runge–Kutta methods

The numerical integration of Hamiltonian systems with oscillatory or periodic solution is considered in this paper. The general framework of constructing trigonometrically fitted symplectic diagonally implicit Runge–Kutta methods is given. A trigonometrically fitted symplectic fourth algebraic order method is constructed. The method is applied to the numerical integration of Hamiltonian systems as the harmonic oscillator, the pendulum and the two body problem and the computation of the eigenvalues of the Schrödinger equation.

Application research of symplectic Runge-Kutta method of solving Lagrange-Maxwell equation

In this paper, we show the numerical integration method of solving Lagrange-Maxwell equation by using the symplectic Runge-Kutta (R-K) method, and numerically study the motion of the plate in an RLC circuit spring coupled system and the current changes. Its result is consistent with that obtained by the traditional R-K method, which demonstrates symplectic integration algorithm is reasonable and effective in studying the electro-mechanical systems. And on this basis, the form invariance of Noether sense is studied by using the symplectic Runge-Kutta method.

Symplectic conditions and stochastic generating functions of stochastic Runge–Kutta methods for stochastic Hamiltonian systems with multiplicative noise

This paper develops numerical approximate algorithms for the solutions of stochastic Hamiltonian systems with multiplicative noise. Some conditions are captured to guarantee that a given stochastic Runge–Kutta method is symplectic. This paper also shows that stochastic symplectic Runge–Kutta methods can be written in terms of stochastic generating functions. Based on the order and symplectic conditions, some low-stage stochastic symplectic Runge–Kutta methods with strong global order 1.0 are constructed. Some numerical results are presented to demonstrate the efficiency of stochastic symplectic Runge–Kutta methods.

Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise

In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM (s,r) methods to construct symplectic Runge–Kutta methods for all values of s and r with s≥r. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge–Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule.

The geometric stricture of the parameter space of the three-stage symplectic Runge-Kutta methods

The geometric structure of the parameter space of the three-stage symplectic Runge-Kutta methods is established. The elements of the structure are geometric places of points corresponding to the methods of different orders of approximation.

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.

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.

Discrete variational calculation of Hénon-Heiles equation in the Birkhoffian sense

The Hénon-Heiles equation that is nonlinear nonintegrable Hamilton system is studied by using the method of discrete variational valculation in the Birkhoffian sense. And the result from the method is compared with the result obtained by using symplectic algorithm and Runge-Kutta method, indicating that the method presented in this paper is reasonable and feasible in computing the dynamical action of the nonlinear nonintegrable Hamilton system in the Birkhoffian sense.

Preservation of equilibria for symplectic methods applied to Hamiltonian systems

In this paper, the linear stability of symplectic methods for Hamiltonian systems is studied. In particular, three classes of symplectic methods are considered: symplectic Runge-Kutta (SRK) methods, symplectic partitioned Runge-Kutta (SPRK) methods and the composition methods based on SRK or SPRK methods. It is shown that the SRK methods and their compositions preserve the ellipticity of equilibrium points unconditionally, whereas the SPRK methods and their compositions have some restrictions on the time-step.

Splitting multisymplectic integrators for Maxwell’s equations

In the paper, we describe a novel kind of multisymplectic method for three-dimensional (3-D) Maxwell’s equations. Splitting the 3-D Maxwell’s equations into three local one-dimensional (LOD) equations, then applying a pair of symplectic Runge–Kutta methods to discretize each resulting LOD equation, it leads to splitting multisymplectic integrators. We say this kind of schemes to be LOD multisymplectic scheme (LOD-MS). The discrete conservation laws, convergence, dispersive relation, dissipation and stability are investigated for the schemes. Theoretical analysis shows that the schemes are unconditionally stable, non-dissipative, and of first order accuracy in time and second order accuracy in space. As a reduction, we also consider the application of LOD-MS to 2-D Maxwell’s equations. Numerical experiments match the theoretical results well. They illustrate that LOD-MS is not only efficient and simple in coding, but also has almost all the nature of multisymplectic integrators.

On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit, symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν=ωh (h is the step size), and for ω→0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255–262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421–434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented.

A sixth-order dual preserving algorithm for the Camassa–Holm equation

The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa–Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge–Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge–Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method.

A new class of symplectic integration schemes based on generating functions

We present a new family of one-step symplectic integration schemes for Hamiltonian systems of the general form $${\dot y=J^{-1}\nabla H(y)^T}$$. Such a class of methods contains as particular cases the methods of Miesbach and Pesch (Numer Math 61:501–521, 1992), and also the family of symplectic Runge-Kutta methods. As in the case of the methods introduced in Miesbach and Pesch (Numer Math 61:501–521, 1992), the new integration methods are constructed by defining a generating function, which automatically determines a symplectic map. The resulting methods are implicit, and require the evaluation of the gradient of the Hamiltonian function as well as the Hessian times a vector.

Modeling and Computation for the High-Speed Rotating Flexible Structure

This paper is presented to improve the modeling accuracy and the computational stability for a high-speed rotating flexible structure. The differential governing equations are derived based on the first-order approximation coupling (FOAC) model theory in the framework of the generalized Hamiltonian principle. The semi-discrete model is obtained by the finite element method, and a new shape function based on FOAC is established for the piezoelectric layers. To increase the efficiency, accuracy, and stability of computation, first, the second-order half-implicit symplectic Runge–Kutta method is presented to keep the computational stability of the numerical simulation in a long period of time. Then, the idea of a precise integration method is introduced into the symplectic geometric algorithm. An improved symplectic precise integration method is developed to increase accuracy and efficiency. Several numerical examples are adopted to show the promise of the modeling and the computational method.

Multi-symplectic Runge–Kutta–Nyström methods for nonlinear Schrödinger equations with variable coefficients

In this paper, we consider Runge–Kutta–Nyström (RKN) methods applied to nonlinear Schrödinger equations with variable coefficients (NLSEvc). Concatenating symplectic Nyström methods in spatial direction and symplectic Runge–Kutta methods in temporal direction for NLSEvc leads to multi-symplectic integrators, i.e. to numerical methods which preserve the multi-symplectic conservation law (MSCL), we present the corresponding discrete version of MSCL. It is shown that the multi-symplectic RKN methods preserve not only the global symplectic structure in time, but also local and global discrete charge conservation laws under periodic boundary conditions. We present a (4-order) multi-symplectic RKN method and use it in numerical simulation of quasi-periodically solitary waves for NLSEvc, and we compare the multi-symplectic RKN method with a non-multi-symplectic RKN method on the errors of numerical solutions, the numerical errors of discrete energy, discrete momentum and discrete charge. The precise conservation of discrete charge under the multi-symplectic RKN discretizations is attested numerically. Some numerical superiorities of the multi-symplectic RKN methods are revealed.

Symplectic Runge-Kutta methods for the Kalman-Bucy filter

Symplectic Runge-Kutta methods for the Kalman-Bucy filter

A New Implementation of Symplectic Runge–Kutta Methods

We propose a “Newton-Taylor” iteration for solving the implicit equations of symplectic Runge-Kutta methods, using the Jacobian of the vector field and matrix-vector multiplications whose extra cost for certain structured problems is negligible. The structure of Hamiltonian ODEs allows this very simple iteration to be effective. The iteration reduces the number of vector field evaluations almost to that of Newton's method, often only one or two per time step, making symplectic Runge-Kutta methods more efficient even at relatively large time steps.

Structure-Preserving Algorithms for a Class of Dynamical Systems

In this paper, we study structure-preserving algorithms for dynamical systems defined by ordinary differential equations in R n . The equations are assumed to be of the form ẏ = A(y)+D(y) +R(y), where A(y) is the conservative part subject to 〈A(y), y〉 = 0; D(y) is the damping part or the part describing the coexistence of damping and expanding; R(y) reflects strange phenomenon of the system. It is shown that the numerical solutions generated by the symplectic Runge-Kutta(SRK) methods with b i > 0 ( i = 1, • • • , s) have long-time approximations to the exact ones, and these methods can describe the structural properties of the quadratic energy for these systems. Some numerical experiments and backward error analysis also show that these methods are better than other methods including the general algebraically stable Runge-Kutta(RK)methods.

Conserved quantities of some Hamiltonian wave equations after full discretization

Hamiltonian PDEs have some invariant quantities, which would be good to conserve with the numerical integration. In this paper, we concentrate on the nonlinear wave and Schrodinger equations. Under hypotheses of regularity and periodicity, we study how a symmetric space discretization makes that the space discretized system also has some invariants or `nearly' invariants which well approximate the continuous ones. We conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic Runge-Kutta methods or symmetric linear multistep methods for second-order systems.

Symplectic structure of poisson system

When the Poisson matrix of Poisson system is non-constant, classical symplectic methods, such as symplectic Runge-Kutta method, generating function method, cannot preserve the Poisson structure. The non-constant Poisson structure was transformed into the symplectic structure by the nonlinear transform. Arbitrary order symplectic method was applied to the transformed Poisson system. The Euler equation of the free rigid body problem was transformed into the symplectic structure and computed by the mid-point scheme. Numerical results show the effectiveness of the nonlinear transform.