On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods

Abstract

Abstract. Differential equations of the formẋ = X = A + B are considered, where the vector fields A and B can be integrated exactly, enabling numerical integration of X by composition of the flows of A and B. Various symmetric compositions are investigated for order, complexity, and reversibility. Free Lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order. A new, more accurate way of applying the methods thus obtained to compositions of an arbitrary first-order integrator is described and tested. The determining equations are explored, and new methods up to 100 times more accurate (at constant work) than those previously known are given. Composition methods.Composition methods are particularly useful for numerically integrating differential equations when the equations have some special structure which it is advantageous to preserve. They tend to have larger local truncation errors than standard (Runge-Kutta, multistep) methods [4,5], but this defect can be more than compensated for by their superior conservation properties.Capital letters such as X will denote vector fields on some space with coordinates x, with flows exp(tX), i.e.,ẋ = X(x) ⇒ x(t) = exp(tX)(x(0)). The vector field X is given and is to be integrated numerically with fixed time step t. Composition methods apply when one can writein such a way that exp(tA), exp(tB) can both be calculated explicitly. Then the most elementary such method is the map (essentially the "Lie-Trotter" formulaThe advantage of composing exact solutions in this way is that many geometric properties of the true flow exp(tX) are preserved: group properties in particular. If X, A, and B are Hamiltonian vector fields then both exp(tX) and the map ϕ 1991 Mathematics Subject Classification. 65L05, 70-08, 58D07, 17B66.

About approximations of exponentials

“…In this paper, we will prove an assertion that has been proposed by R. MacLachlan ( [5]). It shows that the getting of approximations for exp(A 1 + A 2 ) gives approximants for any exp( A i ), as product of first-order approximants.…”

“…We will prove in this paper the assertion proposed by R. MacLachlan ( [5]): The solutions of problems 3 and 2 are equals and there is a one-to-one correspondance between these solutions and the solutions of the problem 1.…”

“…From approximants of exp(A 1 + A 2 ), we deduce approximants for each exp(A 1 +· · ·+A n ), as product of φ + and φ − where φ + is a first order approximant as suggested in ( [5]): φ + (x) = exp(xA 1 ) · · · exp(xA n ).…”

N-body integrators with individual time steps from Hierarchical splitting

“…6: this is the second order leapfrog composition. There is no reason why we could not use a higher order composition scheme (Mclachlan, 1995), as long as we take care selecting the same (or at least of the same order) composition schemes for the component integrators. A problem with using higher order integrators is that the compositions contain terms with negative time steps.…”

Equivariant constrained symplectic integration

“…Thus in this case the constraint algorithm is symmetric, hence second order, automatically. In either case the order can be increased by composition of γ(t i ) and γ −1 (t i ) with appropriately chosen time steps t i [28,31,18].…”