Symplectic Runge-Kutta Methods Research Articles

Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

AbstractWe study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

Dynamic behavior of sail tower SPS based on the symplectic Runge-Kutta method

Space-based solar power as a promising approach to achieve clean and renewable energy had received extensive attention during the last several decades. The dynamics and stability of system of solar power satellite (SPS) in orbit were investigated through the application of Hamilton theory. First, the simplified structure model of sail tower SPS was built, and dynamic governing equation of simplified model of sail tower SPS in the Hamilton form were derived by introducing the concepts of generalized coordinates and generalized momentums and implementing variation method. Then, by numerically solving the equation through the application of the symplectic Runge-Kutta method and comparing the results with the traditional Runge-Kutta method, numerical examples demonstrated that the results achieved by the symplectic Runge-Kutta method indicated the energy preservation as well as the vibrational amplitude of the tether. Finally, the equilibrium points was calculated based on mechanical balance principle. Besides, the stability of the considered system at the equilibrium points was analyzed via numerical examples.

New way to construct high order Hamiltonian variational integrators

This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton’s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and momentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.

Volume preservation by Runge–Kutta methods

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge–Kutta method will respect this property for such systems, but it has been shown by Iserles, Quispel and Tse and independently by Chartier and Murua that no B-Series method can be volume preserving for all volume preserving vector fields. In this paper, we show that despite this result, symplectic Runge–Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discuss how some Runge–Kutta methods can preserve a modified measure exactly.

Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian system. The numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy conservation. Furthermore, they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage

Hamiltonian systems, as one of the most important class of dynamical systems, are associated with a well-known geometric structure called symplecticity. Symplectic numerical algorithms, which preserve such a structure are therefore of interest. In this article, we study the construction of symplectic (partitioned) Runge–Kutta methods with continuous stage. This construction of symplectic methods mainly relies upon the expansion of orthogonal polynomials and the simplifying assumptions for (partitioned) Runge–Kutta type methods. By using suitable quadrature formulae, it also provides a new and simple way to construct symplectic (partitioned) Runge–Kutta methods in classical sense.

Galerkin variational integrators and modified symplectic Runge–Kutta methods

Galerkin variational integrators and modified symplectic Runge–Kutta methods

Symplectic Runge--Kutta Semidiscretization for Stochastic Schrödinger Equation

Based on a variational principle with a stochastic forcing, we indicate that the stochastic Schr\"odinger equation in Stratonovich sense is an infinite-dimensional stochastic Hamiltonian system, whose phase flow preserves symplecticity. We propose a general class of stochastic symplectic Runge-Kutta methods in temporal direction to the stochastic Schr\"odinger equation in Stratonovich sense and show that the methods preserve the charge conservation law. We present a convergence theorem on the relationship between the mean-square convergence order of a semi-discrete method and its local accuracy order. Taking stochastic midpoint scheme as an example of stochastic symplectic Runge-Kutta methods in temporal direction, based on the theorem we show that the mean-square convergence order of the semi-discrete scheme is 1 under appropriate assumptions.

The possibility of constructing a conservative numerical method for the solution of the cauchy problem for Hamiltonian systems based on two-stage symmetric symplectic Runge–Kutta methods

We study the possibility of constructing a computational method for solving the Cauchy problem for Hamiltonian systems, which yields an approximate solution satisfying the law of total energy conservation. The presented technique is based on a family of two-stage symmetric symplectic Runge–Kutta methods. The properties of the proposed method are investigated for the case of a model problem on a material point motion in a cubic field potential. The possibility of working out the method yielding a numerical solution, which preserves the total energy over the period of the problem finite solution except for small neighborhoods of cusps. Time dependences of the symplecticity and reversibility defects are studied on the numerical solution obtained by the constructed method.

Diagonally Implicit Symplectic Runge-Kutta Methods with Special Properties

Diagonally Implicit Symplectic Runge-Kutta Methods with Special Properties

A SIXTH ORDER DIAGONALLY IMPLICIT SYMMETRIC AND SYMPLECTIC RUNGE-KUTTA METHOD FOR SOLVING HAMILTONIAN SYSTEMS

The paper is concerned with construction of symmetric and symplectic Runge-Kutta methods for Hamiltonian systems. Based on the symplectic and symmetrical properties, a sixth-order diagonally implicit symmetric and symplectic Runge-Kutta method with seven stages is presented, the proposed method proved to be P-stable. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing Runge-Kutta methods in scientic literature.

A Low-Dispersive Symplectic Partitioned Runge-Kutta Method for Solving Seismic-Wave Equations: I. Scheme and Theoretical Analysis

Abstract In this article, a series of nearly analytic symplectic partitioned Runge–Kutta (NSPRK) methods for the 3D seismic‐wave equation are developed. First, the Hamiltonian formulations for acoustic and elastic‐wave equations are presented, and then the spatial derivatives are discretized by a nearly analytic discrete operator to obtain a semidiscrete Hamiltonian system. The second‐, third‐, and fourth‐order symplectic partitioned Runge–Kutta schemes are then applied as the time integrator. A semianalytic procedure to facilitate the analysis of stability conditions and numerical dispersion relations is presented, and subsequent theoretical analysis shows that the NSPRK schemes preserve the wave velocity better than conventional symplectic schemes, especially on coarser grids. The numerical solutions computed by NSPRK schemes are compared with analytic solutions for the 3D acoustic and elastic cases. We implemented the NSPRK and some conventional schemes for a 3D acoustic wave propagation simulation in a parallel computer and compared their computational efficiencies. To generate comparable results, the NSPRK schemes require much less computer memory, central processing unit time, and communication time, which substantially accelerates the computation speed. The final simulation in the two‐layer acoustic model shows that the NSPRK schemes can suppress numerical dispersion and preserve the waveforms better than conventional symplectic schemes.

Application of Symplectic Partitioned Runge-Kutta Method and Total-Field Scattered-Field Formulation to Simulation of GPR Data

The second-order Lobatto IIIA-IIIB symplectic partitioned RungeKutta (SPRK) method, combining with the first-order Mur absorbing boundary condition, is developed for the simulation of ground penetrating radar wave propagation in layered pavement structure. For 2-dimetional case, a significant advantage of this method is that only two functions need to be calculated at each time step. The total-field/scattered-field technique is used for plane wave excitation. Numerical examples are presented to verify the accuracy and efficiency of the proposed algorithm. The results illustrate that the reflected signal calculated by the SPRK method is in good agreement with that obtained using the finite difference time domain (FDTD) scheme, but the CPU time consumed by proposed algorithm is reduce about 20% of the FDTD scheme. In addition, an actual field test is conducted to evaluate the further performance of the SPRK method. It is found that the simulated waveform fits well with the measured signal in many aspects, especially in the peak amplitude and time delay.

Diagonally implicit symplectic Runge-Kutta methods with high algebraic and dispersion order.

The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.

Symplectic effective order methods

A family of three stage symplectic Runge–Kutta methods are derived with effective order 4. The methods are constrained so that the coefficient matrix A has only real eigenvalues. This restriction enables transformations to be introduced into the implementation of the method so that, for large Hamiltonian problems, there is a significant gain in efficiency.

Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method

To preserve the symplecticity property, it is natural to require numerical integration of Hamiltonian systems to be symplectic. As a famous numerical integration, it is known that the 2-stage RadauIA method is not symplectic. With the help of symplectic conditions of Runge-Kutta method and partitioned Runge-Kutta method, a symplectic partitioned Runge-Kutta method and a symplectic Runge-Kutta method are constructed on the basis of 2-stage RadauIA method in this paper.

Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method

Symplectic Partitioned Runge-Kutta and Symplectic Runge-Kutta Methods Generated by 2-Stage RadauIA Method

Numerical integration of Hamiltonian problems by G-symplectic methods

It is the purpose of this paper to consider the employ of General Linear Methods (GLMs) as geometric numerical solvers for the treatment of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, we exploit here a concept of near conservation for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve an accurate conservation of the Hamiltonian. In this paper we focus our attention on the connection between order of convergence and Hamiltonian deviation by multivalue methods. Moreover, we derive a semi-implicit GLM which results competitive to symplectic Runge-Kutta methods, which are notoriously implicit.

A unified discontinuous Galerkin framework for time integration

We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time-stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time-stepping schemes, such as low-order one-step methods, Runge-Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge-Kutta methods of different orders are constructed. The derivation of symplectic Runge-Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant-Friedrichs-Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time-stepping schemes.

Runge-kutta method, finite element method, and regular algorithms for hamiltonian system

The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic RungeKutta (RK) method is an important part of the former, and the continuous finite element method (CFEM) belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms—the regular method. Finally, numerical experiments are given to verify the theoretical results.