11th World Congress on Computational Mechanics (WCCM XI)5th European Conference on Computational Mechanics (ECCM V)

6th European Conference on Computational Fluid Dynamics (ECFD VI)

July 20–25, 2014, Barcelona, Spain

STABILITY ANALYSIS OF HIGH ORDER PHASE FITTEDVARIATIONAL INTEGRATORS

Odysseas Kosmas, and Sigrid Leyendecker

Chair of Applied Dynamics, University of Erlangen-Nuremberg, Germanyodysseas.kosmas@ltd.uni-erlangen.de sigrid.leyendecker@ltd.uni-erlangen.de

Key words: variational integrators, discrete variational mechanics, phase lag analysis.

In the present work, we investigate the stability region of high order phase fitted varia-tional integrators using trigonometric interpolation. In the first step, discrete configura-tions and velocities are considered at the nodes of the time grid only (no intermediatepoints are taken into account), see [1, 2].

For the derivation of variational integrators, recalling discrete variational calculus, see [1],the discrete Lagrangian map Ld : Q×Q→ R is defined on two copies of the configurationmanifold Q, which may be considered as an approximation of a continuous action withLagrangian L : TQ→ R, i.e. Ld(qk, qk+1) ≈

∫ tk+1

tkL(q, q)dt in the time interval [tk, tk+1] ⊂

R. The action sum Sd : QN+1 → R, corresponding to the Lagrangian Ld is defined asSd(γd) =

∑N−1k=0 Ld(qk, qk+1) with γd = (q0, . . . , qN) representing the discrete trajectory.

The discrete Hamilton principle states that a motion γd of the discrete mechanical systemextremizes the action sum, i.e. δSd = 0. By differentiation and rearrangement of the termsand having in mind that both q0 and qN are fixed, the discrete Euler-Lagrange equations(DEL) are obtained [1]

D2Ld(qk−1, qk) +D1Ld(qk, qk+1) = 0, k = 1, . . . , N − 1 (1)

where the notation DiLd indicates the slot derivative with respect to the argument of Ld.

Following [3], we restrict ourselves to a linear stability analysis for the harmonic oscillatorproblem with frequency ω and time step h ∈ R, using asymptotically stable numericalsolution uk = (pk, qk), k ∈ N, with p0 = p(0) and q0 = q(0) the initial momentum andposition respectively. The numerical solution uk+1 can be written as

uk+1 = [A(ω, h)]k+1u0 (2)

and the eigenvalues of the amplification matrix A(ω, h) can be further studied [3].

Stability analysis of phase fitted variational integrators

To construct high order methods, we approximate the action integral along the curvesegment between qk and qk+1 using a discrete Lagrangian that depends only on the end

Odysseas Kosmas, and Sigrid Leyendecker

−6 −4 −2 0 2 4 6

10−4e−11

10−2e−11

100

102e−11

104e−11

hω

|λ1,

2|

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.5

0

0.5

1

real(λ1,2

)

imag

(λ1,

2)

Figure 1: Modulus of the eigenvalues of A(ω, h) for the phase fitted variational integrators for S = 2using the eigenvalues of (3) for ωh ∈ [−2π, 2π].

points [1]. We obtain expressions for configurations qjk and velocities qjk for j = 0, ..., S−1,S ∈ N at time tjk ∈ [tk, tk+1] by expressing tjk = tk + Cj

kh for Cjk ∈ [0, 1] such that

C0k = 0, CS−1

k = 1 using qjk = g1(tjk)qk + g2(t

jk)qk+1 and qjk = g1(t

jk)qk + g2(t

jk)qk+1. We

choose functions g1(tjk) = sin

(u− tjk−tk

hu)

(sinu)−1 and g2(tjk) = sin

(tjk−tkhu)

(sinu)−1 to

represent the oscillatory behavior of the solution. For continuity, g1(tk+1) = g2(tk) = 0and g1(tk) = g2(tk+1) = 1 is required [2, 4].

For any choice of interpolation used, we define the discrete Lagrangian by the weightedsum Ld(qk, qk+1) = h

∑S−1j=0 w

jL(q(tjk), q(tjk)), where it can be proved that for maximal

algebraic order,∑S−1

j=0 wj(Cj

k)m = 1

m+1where m = 0, 1, . . . , S − 1 and k = 0, 1, . . . , N − 1

must hold, see [2].

Focussing on the harmonic oscillator, described by the continuous Lagrangian L = q2/2−ω2q2/2, phase fitted variational integrators that use trigonometric interpolating functionswith no intermediate points, i.e. S = 2, can be derived. For those integrators the eigen-values of the amplification matrix in (2) are

λ1,2 =2 cos(2ωh) + 2±

√2 cos(4ωh)− 2

4 cos(ωh). (3)

It can be analytically proved that the phase fitted variational integrator using trigono-metric interpolation for S = 2 is stable for ωh 6= ν π

2and ωh 6= νπ + π

2, ν ∈ Z, i.e. for

the cases that neither the denominator of (3) nor the square root quantities are zero, seeFigure 1. Finally the above procedure can be followed for the linear stability analysis ofphase fitted variational integrator with S > 2.

REFERENCES

[1] J.E. Marsden, and M. West. Discrete mechanics and variational integrators. Acta Nu-merica, Vol. 10, pp. 357-514, 2001.

[2] O.T. Kosmas, and D.S. Vlachos. Phase-fitted discrete Lagrangian integrators. Com-puter Physics Communications, Vol. 181, pp. 562-568, 2010.

[3] B. Leimkuhler and S. Reich. Simulating Hamiltonian Dynamics. Cambridge Univer-sity Press, 2005.

[4] O.T. Kosmas, and S. Leyendecker. Phase lag analysis of variational integrators usinginterpolation techniques. PAMM Proc. Appl. Math. Mech, Vol. 12, pp. 677-678, 2012.

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