Symplecticity of the Störmer-Verlet algorithm for coupling between the shallow water equations and horizontal vehicle motion

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1 Symplectcty of the Störmer-Verlet algorthm for couplng between the shallow water equatons and horzontal vehcle moton by H. Alem Ardakan & T. J. Brdges Department of Mathematcs, Unversty of Surrey, Guldford GU 7XH UK May 5, 00 In ths techncal report t s proved that the Störmer-Verlet algorthm, appled to the dynamcally coupled problem of shallow water sloshng n a horzontally movng vehcle reported n [], s symplectc f the σ ntegral s dscretzed usng the trapezodal rule. The startng pont s the governng equatons for shallow water flud moton n a vehcle constraned to move horzontally wth poston qt. The governng equatons n the Lagrangan partcle path formulaton are [] tt + g χ a a g χ 3 aa + q = 0, a q + νq = ρg L 0 hh d. The frst equaton corresponds to equaton.4 n [] and the second equaton corresponds to equaton. n []. The Hamltonan formulaton of these equatons, wth coordnates qt, a, t, pt, wa, t s gven n 4. of []. The symplectc form s Ω = L Störmer-Verlet dscretzaton 0 dw d ρχ da + dp dq. 0. Consder the Störmer-Verlet dscretzaton for the Hamltonan formulaton as presented n 6 n [], p n+ = p n ν t qn w n+ = w n + g t a 4χ χ n n χ n + n, =,..., N, n+ = n + t w n+ t p n+ + t σ n+, =,..., N, q n+ = q n + t p n+ t σ n+ p n+ = p n+ ν tqn+ w n+ = w n+ + g t a χ 4χ n+ n+ χ n+ + n+, =,..., N,

2 where σ n = L 0 w n ρχ da. The choce of quadrature formula for σ n wll affect symplectcty. At frst we dscretzed usng Smpson s rule, but ths choce does not preserve symplectcty ths wll be apparent n the proof below. We found that the trapezodal rule does ndeed preserve symplectcty. The dscretzed boundary condtons for a, t and wa, t are n+ = 0, n+ N+ = L, w n+/ = w n+/ N+ = p n+ σ n+. The dscretzed symplectc form Usng the trapezodal rule, and notng that δ n = δ N+ = 0, the dscretzaton of the symplectc form 0. s Ω n = ω n + δp n δq n wth ω n := δw n δ n ρχ a.. We say that the numercal scheme s symplectc f Ω n+ = Ω n for all n. 3 Varatonal equatons In order to test for symplectcty, the varatonal equatons assocated wth the dscretzaton are requred. The most complcated ones are the equatons for δw n. The frst one s δw n+ = δw n + g t a χ 4χ δ n n n 3 δ n χ + δ n n + n 3 + δ n, for =,..., N. To smplfy, defne A n = g t a χ n + n 3. Then, for =,..., N, δw n+ = δw n + An χ δ n + An χ δ n An = δ n + t δw n+ t δp n+ + t δσ n+, δw n+ = δw n+/ + An+ χ + An+ χ χ δ n + An χ δ n, An+ χ + An+ χ.

3 The varatonal equatons for q, p are The boundary varatons are δp n+ = δp n ν t δqn δq n+ = δq n + t δp n+ t δσ n+ δp n+ = δp n+ ν t δqn+. δw n+/ = δw n+/ N+ = δp n+ δσ n+ = N+ = The q, p component of the symplectc form δp n+ δq n+ = δp n+ ν tδqn+ δq n+ and so = δp n+ δq n+ = δp n+ δq n t δσ n+ = δp n ν t δqn δq n t δσ n+, δp n+ δq n+ = δp n δq n t δp n δσ n+ + ν t δq n δσ n+. 3. The, w component of the symplectc form From the defntons; for =,..., N, δw n+ = δw n+/ + An+ + χ δn+ + An+ χ δn+. But δw n+/ = δw n+/ δ n t δp n+ + t δσ n+ = δw n δ n + An δ n + δ n + An δ n δ n χ χ t δw n+/ δp n+/ + t δw n+/ δσ n+/. 3

4 Hence Multply by ρχ a δw n+ = δw n δ n + An δ n + δ n + An δ n δ n χ + An+ + χ δn+ ρχ a δw n+ = ρχ a δw n δ n χ + An+ χ δn+ t δw n+/ δp n+/ + t δw n+/ δσ n+/. +ρ aa n δ n + δ n + ρ aa n δ n δ n +ρ aa n+ + δn+ + ρ aa n+ δn+ δn+ t a ρχ δw n+/ δp n+/ + t a ρχ δw n+/ δσ n+/ Proof that the sum of the A n terms vansh Before summng to obtan the symplectc form, frst t s shown that the A n when summed. From the defnton, terms vansh A n δ n + δ n = N A n δ n + δ n = A n δ n δ n =3 usng the fact that δ n N+ = 0, and shftng the nde by one n the second equalty. Smlarly, A n δ n δ n = A n δ n δ n =3 = A n δ n δ n, =3 usng the fact that δ n = 0 and skew-symmetry of wedge. Hence A n δ n + δ n + A n δ n δ n = A smlar argument proves that A n+ + δn+ + A n+ δn+ δn+ =

5 4 Summng to obtan the dscrete symplectc form Snce δ n = 0 and δ n N+ = 0 for all n, we need only sum the terms n 3. from = to = N. Summng and usng 3.3 and 3.4 results n ω n+ = ω n t N ρχ aδw n+/ δp n+/ + t N ρχ aδw n+/ δσ n+/. 4. Ths equaton needs to be combned wth the q, p equaton δp n+ δq n+ = δp n δq n t δp n δσ n+ 5 Quadrature rule for σ n + ν t δq n δσ n+. 4. The terms n parentheses n 4. are partal appromatons of σ n. It s pretty clear that f we use Smpson s rule for σ n, then the dscretzaton wll not be symplectc snce the sum terms n 4. wll not cancel. However, f we use the trapezodal rule then the relevant terms wll cancel. Therefore appromate σ n as σ n = ρ a χ + χ N+ w n + ρχ w n a, usng the fact that w n = wn+ n. The sum n 4. s then N ρχ aδw n+/ = δσ n+ ρ a χ + χ N+ δw n+ 6 Checkng symplectcty Addng the two terms n 4. and 4. Ω n+ = Ω n t. δσ n+ ρ a χ + χ N+ δw n+ δp n+/ + t δσ n+ ρ a χ + χ N+ δw n+ t δp n δσ n+ + ν t δq n δσ n+. Notng that δσ n+ δσ n+ = 0, δp n+ = δp n ν tδqn, and that ths epresson smplfes to provng symplectcty of the scheme. δw n+ = m v δp n+/ δσ n+/, Ω n+ = Ω n, 5 δσ n+/

6 7 Remarks Suppose we start n. wth a dfferent quadrature rule ω n := e δw n δ n ρχ a, where e are weghtngs assocated wth the chosen quadrature formula. Then ths choce wll flter through and σ n can be evaluated wth the same quadrature formula. However, the scheme wll not be symplectc. In ths case the proof breaks down n the proof that the A-terms vansh n 3.3. Unless, the potental term, g χ, a a a s dscretzed dfferently, the symplectc form should be dscretzed usng the trapezodal rule. Although, snce δ n and δ n N+ both vansh, one could equally well use a left or rght Remann sum. Snce w n = wn+ n for all n, the ntegrand n σn wll be perodc when χ = χ N+. Ths s the case for eample when the ntal condton s the flat state, h = h 0. Hence the ntegrand can be etended to all a as a perodc functon. The trapezodal rule s known to have ecellent propertes for perodc functons [3]. However, although the ntegrand s a contnuous perodc functon, t may not be a smooth perodc functon. References [] [] H. Alem Ardakan & T.J. Brdges. Dynamc couplng between shallow-water sloshng and horzontal vehcle moton, Preprnt: Unversty of Surrey 009, avalable electroncally at []. [3] J.A.C. Wedeman. Numercal ntegraton of perodc functons: a few eamples, Amer. Math. Monthly