Six-body problem solution using symplectic integrators

The standard approach to applying IRK methods in the solution of two-point boundary value problems involves the solution of a non-linear system ofn×s equations in order to calculate the stages of the method, wheren is the number of differential equations ands is the number of stages of the implicit Runge-Kutta method. For two-point boundary value problems, we can select a subset of the implicit Runge-Kutta methods that do not require us to solve a non-linear system; the calculation of the stages can be done explicitly, as is the case for explicit Runge-Kutta methods. However, these methods have better stability properties than the explicit Runge-Kutta methods. We have called these new formulas two-point explicit Runge-Kutta (TPERK) methods. Their most important property is that, because their stages can be computed explicity, the solution of a two-point boundary value problem can be computed more efficiently than is possible using an implicit Runge-Kutta method. We have also developed a symmetric subclass of the TPERK methods, called ATPERK methods, which exhibit a number of useful properties.

This paper investigates a class of explicit pseudo three-step Runge-Kutta methods for arbitrarily high order nonstiff initial value problems for systems of first-order differential equations. By using collocation techniques and by suitably choosing collocation points we can obtain a stable \(s\)-stage explicit pseudo three-step Runge-Kutta method (EPThRK method) of order \(p=2s\) requiring only one effective sequential f-evaluation per step on \(s\)-processor parallel computers. By a few widely-used test problems, we show the superiority of the new EPThRK methods proposed in this paper over red well-known parallel PIRK codes and efficient sequential ODEX, DOPRI5 and DOP853 codes available in the literature.

We consider the second order Strang splitting method to approximate the solution to the KdV equation. The model equation is split into three sets of initial value problems containing convection and dispersal terms separately. TVD MUSCL or MUSCL scheme is applied to approximate the convection term and the second order centered difference method to approximate the dispersal term. In time stepping, explicit third order Runge-Kutta method is used to the equation containing convection term and implicit Crank-Nicolson method to the equation containing dispersal term to reduce the CFL restriction. Several numerical examples of weakly and strongly dispersive problems, which produce solitons or dispersive shock waves, or may show instabilities of the solution, are presented.

Due to the chaotic nature of the Solar System, the question of its long-term\nstability can only be answered in a statistical sense, for instance, based on\nnumerical ensemble integrations of nearby orbits. Destabilization of the inner\nplanets, leading to close encounters and/or collisions can be initiated through\na large increase in Mercury's eccentricity, with a currently assumed likelihood\nof ~1%. However, little is known at present about the robustness of this\nnumber. Here I report ensemble integrations of the full equations of motion of\nthe eight planets and Pluto over 5 Gyr, including contributions from general\nrelativity. The results show that different numerical algorithms lead to\nstatistically different results for the evolution of Mercury's eccentricity\n(eM). For instance, starting at present initial conditions (eM ~= 0.21),\nMercury's maximum eccentricity achieved over 5 Gyr is on average significantly\nhigher in symplectic ensemble integrations using heliocentricthan Jacobi\ncoordinates and stricter error control. In contrast, starting at a possible\nfuture configuration (eM ~= 0.53), Mercury's maximum eccentricity achieved over\nthe subsequent 500 Myr is on average significantly lower using heliocentric\nthan Jacobi coordinates. For example, the probability for eM to increase beyond\n0.53 over 500 Myr is >90% (Jacobi) vs. only 40-55% (heliocentric). This poses a\ndilemma as the physical evolution of the real system - and its probabilistic\nbehavior - cannot depend on the coordinate system or numerical algorithm chosen\nto describe it. Some tests of the numerical algorithms suggest that symplectic\nintegrators using heliocentric coordinates underestimate the odds for\ndestabilization of Mercury's orbit at high initial eM.\n

We present an algorithm for determining the stepsize in an explicit Runge-Kutta method that is suitable when solving moderately stiff differential equations. The algorithm has a geometric character, and is based on a pair of semicircles that enclose the boundary of the stability region in the left half of the complex plane. The algorithm includes an error control device. We describe a vectorized form of the algorithm, and present a corresponding MATLAB code. Numerical examples for Runge-Kutta methods of third and fourth order demonstrate the properties and capabilities of the algorithm.

This paper is aimed at analyzing the consistency and convergence of a derived explicit fourth-stage fourth-order Runge-Kutta method. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.

In this paper we present new modified open Newton Cotes integrators and we develop a new modified trigonometrically-fitted open Newton-Cotes method. We study the connection between the new proposed schemes, the differential methods and the symplectic integrators. although The research on multistep symplectic integrators is very poor, although, much research has been done on one step symplectic integrators and several of then have obtained based on symplectic geometry. In this paper a new open modified numerical algorithm of Newton-Cotes type is produced. We present the new obtained method as symplectic multilayer integrator. The new obtained symplectic schemes are applied for the solution of the resonance problem of the radial Schrodinger Equation. The results show the efficiency of the new proposed algorithm.

The dynamics of spacecraft relative motion greatly counts in the planning, guidance and control of many space missions, such as on-orbit servicing and formation flight. This paper develops a Hamiltonian model for spacecraft relative motion in the framework of analytical mechanics. Considering the structure-conserved property of Hamiltonian systems and deficiency of the classical explicit fourth-order Runge-Kutta method, the symplectic Runge-Kutta method is utilized to propagate the Hamiltonian model. Additionally, the applicability of the Hamiltonian model is numerically analyzed via a comparison with the well-known Clohessy-Wiltshire equations and Tschauner-Hempel equations.

In this chapter, the discretization of continuous systems is presented. The explicit and implicit discrete maps are discussed for numerical predictions of continuous systems. Basic discrete schemes are presented which include forward and backward Euler methods, midpoint, and trapezoidal rule method. An introduction to Runge–Kutta methods is presented, and the Taylor series method and second-order Runge–Kutta method are introduced. The explicit Runge–Kutta methods for third and fourth order are systematically presented. The implicit Runge–Kutta methods are discussed based on the polynomial interpolation, which include a generalized implicit Runge–Kutta method, Gauss method, Radau method, and Lotta methods. In addition to one-step methods, implicit and explicit multi-step methods are discussed, including Adams–Bashforth method, Adams–Moulton methods, and explicit and implicit Adams methods.

Variational formulations for explicit Runge-Kutta Methods

Explicit Runge–Kutta methods and implicit multistep methods utilizing a Newton–Krylov nonlinear solver are evaluated for a range of configurations of the shallow-water dynamical core of the spectral element community atmosphere model to evaluate their computational performance. These configurations are designed to explore the attributes of each method under different but relevant model usage scenarios including varied spectral order within an element, static regional refinement, and scaling to the largest problem sizes. This analysis is performed within the shallow-water dynamical core option of a full climate model code base to enable a wealth of simulations for study, with the aim of informing solver development within the more complete hydrostatic dynamical core used for climate research. The limitations and benefits to using explicit versus implicit methods, with different parameters and settings, are discussed in light of the trade-offs with Message Passing Interface (MPI) communication and memory and their inherent efficiency bottlenecks. Given the performance behavior across the configurations analyzed here, the recommendation for future work using the implicit solvers is conditional based on scale separation and the stiffness of the problem. For the regionally refined configurations, the implicit method has about the same efficiency as the explicit method, without considering efficiency gains from a preconditioner. The potential for improvement using a preconditioner is greatest for higher spectral order configurations, where more work is shifted to the linear solver. Initial simulations with OpenACC directives to utilize a Graphics Processing Unit (GPU) when performing function evaluations show improvements locally, and that overall gains are possible with adjustments to data exchanges.

Computer assisted derivation and improved techniques have led to effective explicit Runge-Kutta methods of higher order. These methods become inefficient when the step size must be reduced often to produce approximations at specified points. Considerable effort has been devoted to providing Runge-Kutta methods with an interpolation capability, so that approximations can be produced inexpensively at intermediate points of a successful step. New high order Hermite interpolants for two well known embedded Runge-Kutta methods of orders 7 and 8 are presented. These interpolants are constructed using values from two successive integration steps, are locally of O(h 8) or O(h 9), and require only one or four extra function evaluations per step respectively.

We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the differential form of EERK method by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an EERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. A simple indicator, namely average dissipation rate, is also introduced for these multi-stage methods to evaluate the overall energy dissipation rate of an EERK method such that one can choose proper parameters in some parameterized EERK methods or compare different kinds of EERK methods. Some existing EERK methods in the literature are evaluated from the perspective of preserving the original energy dissipation law and the energy dissipation rate. Some numerical examples are also included to support our theory.

In this paper we study a class of explicit pseudo two-step Runge-Kutta (EPTRK) methods for first-order ODEs for parallel computers. We investigate linear stability and derive methods with enlarged stability regions. In numerical experiments on a shared memory computer we compare a parallel variable step size EPTRK implementation with the effcient sequential Runge-Kutta method dopri5.

Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer