Symplectic Runge-Kutta Research Articles

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 Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More

It is well known that symplectic Runge-Kutta and Partitioned Runge-Kutta methods exactly preserve {\em quadratic} first integrals (invariants of motion) of the system being integrated. While this property is often seen as a mere curiosity (it does not hold for arbitrary first integrals), it plays an important role in the computation of numerical sensitivities, optimal control theory and Lagrangian mechanics, as described in this paper, which, together with some new material, presents in a unified way a number of results now scattered or implicit in the literature. Some widely used procedures, such as the direct method in optimal control theory and the computation of sensitivities via reverse accumulation imply "hidden" integrations with symplectic Partitioned Runge-Kutta schemes.

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.

Simulation of Incompressible Free Surface Flow Using the Volume Preserving Level Set Method

AbstractThis study aims to develop a numerical scheme in collocated Cartesian grids to solve the level set equation together with the incompressible two-phase flow equations. A seventh-order accurate upwinding combined compact difference (UCCD7) scheme has been developed for the approximation of the first-order spatial derivative terms shown in the level set equation. Developed scheme has a higher accuracy with a three-point grid stencil to minimize phase error. To preserve the mass of each phase all the time, the temporal derivative term in the level set equation is approximated by the sixth-order accurate symplectic Runge-Kutta (SRK6) scheme. All the simulated results for the dam-break, Rayleigh-Taylor instability, bubble rising, two-bubble merging, and milkcrown problems in two and three dimensions agree well with the available numerical or experimental results.

Construction of Exponentially Fitted Symplectic Runge–Kutta–Nyström Methods from Partitioned Runge–Kutta Methods

In this work, we give the general framework for constructing trigonometrically fitted symplectic Runge–Kutta–Nystrom (RKN) methods from symplectic trigonometrically fitted partitioned Runge–Kutta (PRK) methods. We construct RKN methods from PRK methods with up to five stages and fourth algebraic order. Numerical results are given for the two-body problem and the perturbed two-body problem.

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.

Explicit symplectic RKN methods for perturbed non-autonomous oscillators: Splitting, extended and exponentially fitting methods

We consider the numerical integration of perturbed non-autonomous oscillatory systems using high order methods. The autonomous case has been efficiently integrated using explicit and symplectic Runge–Kutta–Nyström (RKN) methods like extended RKN methods, exponentially fitting RKN methods and splitting methods for perturbed systems. Recently, it has been shown that explicit and symplectic extended RKN methods and exponentially fitting RKN methods are equivalent (Wu et al., 2012) and in this work we show that these methods are also equivalent to splitting methods for perturbed oscillators. We provide a constructive proof which at the same time allows us to build for the first time new explicit and symplectic extended RKN methods for the non-autonomous problem (for multidimensional time dependent frequencies). The new methods obtained, while built from splitting methods, are different in the treatment of the time-dependent terms and can be superior in some cases. We build some new methods and show their performance on numerical examples.

A Low‐Dispersive Symplectic Partitioned Runge–Kutta Method for Solving Seismic‐Wave Equations: II. Wavefield Simulations

In a companion paper (Ma et al. , 2014), we proposed a series of nearly analytic symplectic Runge–Kutta (NSPRK) methods with fourth‐order spatial accuracy and with second‐, third‐, and fourth‐order temporal accuracy. In this article, the NSPRK(2,4) scheme, which is temporally second order and the most efficient of the NSPRK schemes, is applied to 3D seismic modeling. We apply the NSPRK(2,4) scheme to simulate acoustic or elastic wave propagation in common 3D medium models, including a three‐layer acoustic model and a vertical transversely isotropic model. A model with a fluid‐filled fracture is used to study elastic wave propagation. When it was necessary to truncate the computational domain, we used the classical split‐field perfectly matched layer (PML) approach to eliminate reflected waves. Compared with conventional methods, the NSPRK method requires only a little additional computation and memory storage on PML domains to obtain a large absorption effect. Numerical results show no visible numerical dispersion, although all simulations were done on relatively coarse grids. These promising numerical results support the use of NSPRK‐type schemes on large‐scale practical models. The NSPRK method with symplecticity‐preserving properties can provide more accurate wave amplitudes than other methods. Online Material: MATLAB code.

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.

Fast Ion Trajectory Calculations in Tokamak Magnetic Configuration Using Symplectic Integration Algorithm

A numerical scheme based on Symplectic Integration Algorithm (SIA) has been used to develop an orbit following code to calculate fast ion trajectories in tokamak magnetic configuration. For the purpose of demonstrating the expediency of symplectic schemes, the algorithm has been applied to the H´enon-Heiles system and compared with non- symplectic Runge–Kutta Algorithm (RKA) for numerically integrating the Hamiltonian equations. In contrast to RKA, the long-time stability of SIA has been highlighted. Furthermore, the SIA has been used to find the exact trajectories of, trapped and passing, fast ions in tokamak. In particular, the effect of an intrinsic magnetic field perturbation has been investigated, i.e. toroidal field ripples. This perturbation is toroidal field ripple (TFR) which are there due the discrete number to toroidal coils around the torus. The numerical scheme used shows excellent conservation of particle energy as well as of angular momentum (in axi-symmetric case). The effect of TFR on these trajectories has been simulated and it is shown that the resonance between toroidal precession of bananas and the field ripples results in spread of the trajectories for banana particles, whereas the passing fast ions are unaffected.

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.

A fourth order modified trigonometrically fitted symplectic Runge–Kutta–Nyström method

In this work we construct a modified trigonometrically fitted symplectic Runge Kutta Nyström method based on the fourth order five stages method of Calvo and Sanz-Serna (1994). We apply the new method on the numerical integration of the two-dimensional harmonic oscillator, the two-body problem, a perturbed two-body problem and two two-dimensional nonlinear oscillatory Hamiltonian systems.

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.

Collective Lie-Poisson integrators on R3

We develop Lie-Poisson integrators for general Hamiltonian systems on $\mathbf{R}^{3}$ equipped with the rigid body bracket. The method uses symplectic realisation of $\mathbf{R}^{3}$ on $T^{*}\mathbf{R}^{2}$ and application of symplectic Runge-Kutta schemes. As a side product, we obtain simple symplectic integrators for general Hamiltonian systems on the sphere $S^{2}$.

Canonicalization and symplectic simulation of the gyrocenter dynamics in time-independent magnetic fields

The gyrocenter dynamics of charged particles in time-independent magnetic fields is a non-canonical Hamiltonian system. The canonical description of the gyrocenter has both theoretical and practical importance. We provide a general procedure of the gyrocenter canonicalization, which is expressed by the series of a small variable ϵ depending only on the parallel velocity u and can be expressed in a recursive manner. We prove that the truncation of the series to any given order generates a set of exact canonical coordinates for a system, whose Lagrangian approximates to that of the original gyrocenter system in the same order. If flux surfaces exist for the magnetic field, the series stops simply at the second order and an exact canonical form of the gyrocenter system is obtained. With the canonicalization schemes, the canonical symplectic simulation of gyrocenter dynamics is realized for the first time. The canonical symplectic algorithm has the advantage of good conservation properties and long-term numerical accuracy, while avoiding numerical instability. It is worth mentioning that explicitly expressing the canonical Hamiltonian in new coordinates is usually difficult and impractical. We give an iteration procedure that is easy to implement in the original coordinates associated with the coordinate transformation. This is crucial for modern large-scale simulation studies in plasma physics. The dynamics of gyrocenters in the dipole magnetic field and in the toroidal geometry are simulated using the canonical symplectic algorithm by comparison with the higher-order non symplectic Runge-Kutta scheme. The overwhelming superiorities of the symplectic method for the gyrocenter system are evidently exhibited.

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

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