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Symplectic Phase Flow Approximation for the Numerical Integration of Canonical Systems S. Miesbach and H. J. Pesch
Mathematisches Institut, TU Munchen, Arcisstrae 21, D-8000 Munchen 2, Federal Republic of Germany, Telefax (+49-89) 2105-8156
Summary. New methods are presented for the numerical integration of ordinary
di erential equations of the important family of Hamiltonian dynamical systems. These methods preserve the Poincare invariants and, therefore, mimic relevant qualitative properties of the exact solutions. The methods are based on a RungeKutta-type ansatz for the generating function to realize the integration steps by canonical transformations. A fourth-order method is given and its implementation is discussed. Numerical results are presented for the Henon-Heiles system, which describes the motion of a star in an axisymmetric galaxy. Subject Classi cations: AMS(MOS): 65L05, 65L07, 58F05, 70-08, 70F15, 70H15; CR: G1.7
1 Introduction Numerical methods for the solution of initial-value problems of ordinary di erential equations (ODEs) have achieved high reliability and eciency as well as wide applicability. Obviously, generality may have certain drawbacks since a \general" method cannot be optimal in all individual cases. The objective of the present paper is to consider numerical methods for special ODEs which can be described by so-called Hamiltonian or canonical systems. Such systems of ODEs frequently arise in mathematical physics, e.g., in the description of nondissipative problems.
2 An autonomous Hamiltonian system is determined by a real valued function H (p; q) over the phase space P := 2n = fz := (p; q)j p = (p1 ; . . . ; pn ); q = (q1 ; . . . ; qn )g where n is the degree of freedom. The time evolution of the system is given by the following ODE system IR
(1:1)
@H ; = ? p_ = dp dt @q
@H q_ = dq = dt @p
or, in short, (1:2)
z_ = J ?1 @H @z
where (1:3)
0 I J= ?I 0
with the n n identity matrix I . The Hamiltonian function H usually corresponds to the physical energy of the system. The variables pi and qi denote the generalized momentum and spatial coordinates, respectively. Let the evolution operator gt map each point z 2 P , regarded as initial value z := z(0) , to the point gt (z) = z(t) on the corresponding trajectory through z . This operator is called the Hamiltonian phase ow. Indeed, gt is a ow since there holds, due to autonomy, (1:4)
gt gs = gt+s
for all times t and s for which the operators in (1.4) are de ned. If one disregards the occurrence of singularities, the set fgtj t 2 g becomes a one parameter group with identity g0 and inverse (gt)?1 = g?t . For more details see Arnol'd [1]. Considering the phase ow on a discrete time grid tj = j h with j 2 and a xed stepsize h , the evolution becomes an iteration process IR
Z Z
(1:5)
z(tj+1 ) = gh(z(tj )) :
Explicit and implicit numerical one-step integration methods with constant stepsize h also t into that pattern, (1:6)
zj+1 = Fh (zj ) :
Here zj denotes the approximation of z(tj ) , and Fh describes the application of one integration step of the numerical scheme. Thus, Fh can be interpreted as a phase ow approximation.
3 Especially for meaningful long time simulations it is necessary to construct phase ow approximations in such a way that intrinsic physical properties of the underlying canonical system are preserved. One of the most important properties are the so-called symplectic invariants. For each degree of freedom the Hamiltonian ow gt satis es a 2i-dimensional area preserving law (for i = 1; . . . ; n ) known as the Poincare-Cartan integral invariant. The n-th invariant is equivalent to the preservation of the phase volume (see Arnol'd [1]). Generally, a transformation ' on P preserving these symplectic invariants is called canonical. Its Jacobian @' @z has to satisfy > J ( @' ) = J : ) ( @' @z @z
(1:7)
This paper is concerned with numerical integration schemes Fh which satisfy the symplectic preservation laws like the underlying Hamiltonian ow. Therefore, the Jacobian of Fh has also to satisfy (1.7). The construction of canonical integration methods is the aim of the present paper. For the bene t of the reader, the following example of a harmonic oscillator should illustrate the gain in meaningfulness of the numerical results when using symplectic instead of nonsymplectic methods. The Hamiltonian of the oscillator is given by
H (p; q) = 21 (p2 + k2 q2) :
(1:8)
The Hamiltonian phase ow gt is the linear transformation (1:9)
p(t) = q(t)
|
cos kt ?k sin kt 1 sin kt cos kt k {z =: Gt
}
p : q
In two dimensions the canonicity condition (1.7) reduces to det Gt = 1 . Applying Heun's method (see, e.g., Stoer and Bulirsch [20], p. 414) ? zj+1 = zj + h2 f (zj ) + f (zj + hf (zj ))
(1:10)
for an ODE with right-hand side f to the harmonic oscillator, one obtains (1:11)
2 pj+1 = 1 ? h2 k2 ?hk22 qj+1 h {z1 ? h2 k2 | =: FhHeun
}
pj qj
:
4 The centered Euler method (CEM) j j +1 zj+1 = zj + hf ( z +2z )
(1:12) leads to (1:13)
1 1 ? h42 k2 ?hk2 pj+1 = qj+1 h 1 ? h42 k2 1 + h42 k2 | {z =: FhCEM
}
pj qj
:
Checking the canonicity condition, one sees that Heun's method is nonsymplectic, i.e., det FhHeun > 1 , whereas the centered Euler method is symplectic, i.e., det FhCEM = 1 for all stepsizes h 2 . IR
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