The Control of Parasitism in $G$-symplectic Methods

John C. Butcher, Yousaf Habib, Adrian T. Hill, Terence J. T. Norton
2014 SIAM Journal on Numerical Analysis  
G-symplectic general linear methods are designed to approximately preserve symplectic invariants for Hamiltonian systems. In this paper, the properties of G-symplectic methods are explored computationally and theoretically. Good preservation properties are observed over long times for many parameter ranges, but, for other parameter values, the parasitic behavior, to which multivalue methods are prone, corrupts the numerical solution by the growth of small perturbations. Two approaches for
more » ... pproaches for alleviating this effect are considered. First, compositions of methods with growth parameters of opposite signs can be used to cancel the long-term effect of parasitism. Second, methods can be constructed for which the growth parameters are zero by design. Each of these remedies is found to be successful in eliminating parasitic behavior in long-term simulations using a variety of test problems. PARASITISM IN G-SYMPLECTIC METHODS 2441 symplectic Euler and Runge-Kutta-Nyström methods have been generalized to obtain higher order partitioned Runge-Kutta methods [15] , some of which are explicit. The most popular low order method for separable problems is the explicit Verlet method [26] , which may be viewed as a partitioned Runge-Kutta method, a partitioned linear multistep method, or a nonstandard implementation of the leapfrog method. Conservation properties of standard linear multistep methods were investigated by Eirola and Sanz-Serna [10], who showed that all time-symmetric methods have the G-symplectic property. Although dimensional incommensurability implies that such methods cannot be symplectic themselves, Hairer [12] showed that the underlying one-step method of a symmetric linear multistep method posesses a property similar to symplecticity using a general result on conjugate symplecticity proved in [7] . A related question for G-symplectic general linear methods is whether their underlying one-step method is also conjugate symplectic. Although this question remains open theoretically, analogy with the linear multistep case and the computational evidence of section 2.3 suggest that the answer is affirmative. Unfortunately for multivalue methods, a well-behaved underlying one-step method is insufficient to ensure that the method is effective. A second requirement is that the growth of parasitic components is controlled. However, G-symplecticity implies that the parasitic eigenvalues ζ j (z) are unimodular for z = 0. Indeed, for the case of linear multistep methods, the parasitic growth parameters μ j = σ(ζ j )/(ζ j ρ (ζ j )) [8] are nonzero [14, Chapter XIV], leading to numerical instability for many problems of interest. Thus, standard linear symmetric multistep methods have limited applicability. More promisingly, in the case of s-stable time-symmetric linear multistep methods for second order differential equations [13], backward error analysis techniques have been used to show that parasitic components remain under control over long times, so that the numerical solution is largely determined by the well-behaved principal component of the solution. Higher order versions of these methods are applied and analyzed in [19, 15] . G-symplectic general linear methods face hazards from their parasitic components similar to those encountered by standard linear multistep methods. In this paper we consider two distinct approaches to controlling parasitism. First, for the case of a two-step method for which there is one nonzero parasitic component, we compose two numerical methods with growth parameters of opposite signs in such a way that the overall effects cancel out over a long time. This approach is applied to two related fourth order methods. Second, we construct G-symplectic general linear methods for which all parasitic growth parameters are zero. The effectiveness of these two approaches in approximately preserving energy and symplectic invariants over long times is demonstrated by a number of computations. The new methods constructed here are of up to order 4 and diagonally implicit with, in some cases, extra zeros on the diagonal. As they are designed for general Hamiltonian problems, they are in direct competition with symplectic DIRKs. Their construction and analysis depends on simplifications and developments in Diagonally Implicit Multistage Integration method (DIMSIM) general linear methods over the past two decades, especially methods designed for stiff dissipative problems [2, 6, 3]. The paper is organized as follows. In section 2, G-symplectic general linear methods are introduced, and this is followed in section 3 by an account of their parasitic behavior. In section 4, composition of methods is considered as an approach to overcoming the damaging effects of parasitism. In section 5, methods are constructed Downloaded 11/11/15 to Redistribution subject to SIAM license or copyright; see
doi:10.1137/140953277 fatcat:xgjq6ph2sfdltks4rvykjrdxpi