Chaper Numerical Tess We here collec a few numerical ess, in order o pu ino evidence he poenialiies of HBVMs [4, 6, 7]. Tes problem Le us consider he problem characerized by he polynomial Hamilonian (4.) in [9], H(p, q) = p3 3 p + q6 3 + q4 4 q3 3 + 6, (.) havingdegree ν = 6, saringa he iniial poin y (q(), p()) T = (, ) T, so ha H(y ) =. For such a problem, in [9] i has been experienced a numerical drif in he discree Hamilonian, when usinghe fourh-order Lobao IIIA mehod wih sepsize h =.6, as confirmed by he plo in Figure.. When using he fourh-order Gauss-Legendre mehod he drif disappears, even hough he Hamilonian is no exacly preserved alonghe discree soluion, as is confirmed by he plo in Figure.. On he oher hand, by usinghe fourh-order HBVM(6,) wih he same sepsize, he Hamilonian urns ou o be preserved up o machine precision, as shown in Figure.3, since such mehod exacly preserves polynomial Hamilonians of degree up o 6. In such a case, accordingo he las iem in Remark7, he numerical soluions obained by usinghe Lobao nodes {c =, c,..., c 6 = } or he Gauss-Legendre nodes {c,..., c 6 } are he same. The fourh-order convergence of he mehod is numerically verified by he resuls lised in Table.. 5
6 CHAPTER. NUMERICAL TESTS x 6 H 3 4 5 5 5 Figure.:Fourh-order Lobao IIIA mehod, h =.6, problem (.):drif in he Hamilonian. 3.5 x 6 3.5 H.5.5.5 5 5 Figure.:Fourh-order Gauss-Legendre mehod, h =.6, problem (.):H 6. x 6 H 3 4 5 5 Figure.3:Fourh-order HBVM(6,) mehod, h =.6, problem (.):H 6.
7 Tes problem The second es problem, havinga highly oscillaingsoluion, is he Fermi-Pasa- Ulam problem (see [, Secion I.5.]), modellinga chain of m mass poins conneced wih alernaingsof nonlinear and siff linear springs, and fixed a he end poins. The variables q,..., q m sand for he displacemens of he mass poins, and p i = q i for heir velociies. The correspondinghamilonian, represeninghe oal energy, is H(p, q) = m ( p i + pi) ω + i= 4 m m (q i q i ) + (q i+ q i ) 4, (.) i= i= wih q = q m+ =. In our simulaion we have used he followingvalues: m = 3, ω = 5, and saringvecor p i =, q i = (i )/, i =,..., 6. In such a case, he Hamilonian funcion is a polynomial of degree 4, so ha he fourh-order HBVM(4,) mehod, eiher when usinghe Lobao nodes or he Gauss-Legendre nodes, is able o exacly preserve he Hamilonian, as confirmed by he plo in Figure.6, obained wih sepsize h =.5. Conversely, by using he same sepsize, boh he fourh-order Lobao IIIA and Gauss-Legendre mehods provide only an approximae conservaion of he Hamilonian, as shown in he plos in Figures.4 and.5, respecively. The fourh-order convergence of he HBVM(4,) mehod is numerically verified by he resuls lised in Table..
8 CHAPTER. NUMERICAL TESTS x 4 8 6 H H 4 4 6 3 4 5 Figure.4:Fourh-order Lobao IIIA mehod, h =.5, problem (.): H H 3. 4 x 4 H H 4 6 8 3 4 5 Figure.5:Fourh-order Gauss-Legendre mehod, h =.5, problem (.): H H 3..5 x 4.5 H H.5.5 3 4 5 Figure.6:Fourh-order HBVM(4,) mehod, h =.5, problem (.): H H 4.
9 Tes problem 3(non-polynomial Hamilonian) In he previous examples, he Hamilonian funcion was a polynomial. Neverheless, as observed above, also in his case HBVM(k,s) are expeced o produce a pracical conservaion of he energy when applied o sysems defined by a nonpolynomial Hamilonian funcion ha can be locally well approximaed by a polynomial. As an example, we consider he moion of a charged paricle in a magneic field wih Bio-Savar poenial. I is defined by he Hamilonian [6] H(x, y, z, ẋ, ẏ, ż) = (.3) [ ( ẋ α x ) + (ẏ α yϱ ) ] + (ż + α log(ϱ)), m ϱ wih ϱ = x + y, α = e B, m is he paricle mass, e is is charge, and B is he magneic field inensiy. We have used he values wih saringpoin m =, e =, B =, x =.5, y =, z =, ẋ =., ẏ =.3, ż =. By usinghe fourh-order Lobao IIIA mehod, wih sepsize h =., a drif is again experienced in he numerical soluion, as is shown in Figure.7. By usinghe fourh-order Gauss-Legendre mehod wih he same sepsize, he drif disappears even hough, as shown in Figure.8, he value of he Hamilonian is preserved wihin an error of he order of 3. On he oher hand, when using he HBVM(6,) mehod wih he same sepsize, he error in he Hamilonian decreases o an order of 5 (see Figure.9), hus givinga pracical conservaion. Finally, in Table.4 we lis he maximum absolue difference beween he numerical soluions over 3 inegraion seps, compued by he HBVM(k, ) mehods based on Lobao abscissae and on Gauss-Legendre abscissae, as k grows, wih sepsize h =.. We observe ha he difference ends o, as k increases. Finally, also in his case, one verifies a fourh-order convergence, as he resuls lised in Table.3show. This kind of moion causes he well known phenomenon of aurora borealis.
CHAPTER. NUMERICAL TESTS 6 x 3 4 H H 4 6 8 5 5 Figure.7:Fourh-order Lobao IIIA mehod, h =., problem (.3):drif in he Hamilonian. 5 x 3 H H 5 5 5 Figure.8:Fourh-order Gauss-Legendre mehod, h =., problem (.3): H H 3. x 5 H H 3 5 5 Figure.9:Fourh-order HBVM(6,) mehod, h =., problem (.3): H H 5.
Table.:Numerical order of convergence for he HBVM(6,) mehod, problem (.). h.3.6.8.4. error.88.487 3 9.398 5 5.89 6 3.684 7 order 3.94 3.98 4. 4. Table.:Numerical order of convergence for he HBVM(4,) mehod, problem (.). h.6 8 3 4 3 3 3 error 3.3.967.4 7.76 4 4.853 5 order 3.97 3.99 4. 4. Table.3:Numerical order of convergence for he HBVM(6,) mehod, problem (.3). h 3..6 8 3 4 3 3 error 3.944 6.635 7.79 8.94 9 6.838 order 3.9 3.93 3.98 4. Table.4:Maximum difference beween he numerical soluions obained hrough he fourh-order HBVM(k, ) mehods based on Lobao abscissae and Gauss- Legendre abscissae for increasingvalues of k, problem (.3), 3 seps wih sepsize h =.. k h =. 3.97 4.9 3 6. 8 8.37 5.88 3
CHAPTER. NUMERICAL TESTS Tes problem 4(Sinikovproblem) The main problem in Celesial Mechanics is he so called N-body problem, i.e. o describe he moion of N poin paricles of posiive mass movingunder Newon s law of graviaion when we know heir posiions and velociies a a given ime. This problem is described by he Hamilonian funcion: H(q, p) = N i= p i m i G N i= i m j m i, (.4) q i q j j= where q i is he posiion of he ih paricle, wih mass m i, and p i is is momenum. The Sinikov problem is a paricular configuraion of he 3-body dynamics (see, e.g., [3]). In his problem wo bodies of equal mass (primaries) revolve abou heir cener of mass, here assumed a he origin, in ellipic orbis in he xyplane. A hird, and much smaller body (planeoid), is placed on he z-axis wih iniial velociy parallel o his axis as well. The hird body is small enough ha he wo body dynamics of he primaries is no desroyed. Then, he moion of he hird body will be resriced o he z-axis and oscillaingaround he origin bu no necessarily periodic. In fac his problem has been shown o exhibi a chaoic behavior when he eccenriciy of he orbis of he primaries exceeds a criical value ha, for he daa se we have used, is ē.75 (see Figure.). We have solved he problem defined by he Hamilonian funcion (.4) by he Gauss mehod of order 4 (i.e., HBVM(,) a Gaussian nodes) and by HBVM(8,) a 8 Gaussian nodes (order 4, fundamenal and 6 silen sages), wih he followingse of parameers in (.4): N G m m m 3 e d h max 3 5.75 5.5 5 where e is he eccenriciy, d is he disance of he apocenres of he primaries (poins a which he wo bodies are he furhes), h is he used ime-sep, and [, max] is he ime inegraion inerval. The eccenriciy e and he disance d may be used o define he iniial condiion [q, p ] (see [3] for he deails): q = [ 5,,, 5,,,,, 9 ] T, p = [,,,,,,,, ]T. Firs of all, we consider he wo picures in Figure. reporinghe relaive errors in he Hamilonian funcion and in he angular momenum evaluaed alonghe numerical soluions compued by he wo mehods. We know ha he
HBVM(8,) precisely conserves Hamilonian polynomial funcions of degree a mos 8. This accuracy is high enough o guaranee ha he nonlinear Hamilonian funcion (.4) is as well conserved up o he machine precision (see he upper picure):from a geomerical poin of view his means ha a local approximaion of he level curves of (.4) by a polynomial of degree 8 leads o a negligible error. The Gauss mehod exhibis a cerain error in he Hamilonian funcion while, beinghis formula symplecic, i precisely conserves he angular momenum, as is confirmed by lookinga he down picure of Figure.. The error in he numerical angular momenum associaed wih he HBVM(8,) undergoes some bounded periodic-like oscillaions. Figures. and.3show he numerical soluion compued by he Gauss mehod and HBVM(8,), respecively. Since he mehods leave he xy-plane invarian for he moion of he primaries and he z-axis invarian for he moion of he planeoid, we have jus repored he moion of he primaries in he xy-phase plane (upper picures) and he space-ime diagram of he planeoid (down picure). We observe ha, for he Gauss mehod, he orbis of he primaries are irregular in characer so ha he hird body, afer performingsome oscillaions around he origin, will evenually escape he sysem (see he down picure of Figure.). On he conrary (see he upper picure of Figure.3), he HBVM(8,) mehod generaes a quie regular phase porrai. Due o he large sepsize h used, a sham roaion of he xy-plane appears which, however, does no desroy he global symmery of he dynamics, as esified by he bounded oscillaions of he planeoid (down picure of Figure.3) which lookvery similar o he reference ones in Figure.. This aspec is also confirmed by he picures in Figure.4 displayinghe disance of he primaries as a funcion of he ime. We see ha he disance of he apocenres (correspondingo he maxima in he plos), as he wo bodies wheel around he origin, are preserved by he HBVM(8,) (down picure) while he same is no rue for he Gauss mehod (upper picure). 3
4 CHAPTER. NUMERICAL TESTS.5.5 3 3.5.5.5.5 5 5 5 3 35 Figure.:The upper picure displays he configuraion of 3-bodies in he Sinikov problem. To an eccenriciy of he orbis of he primaries e =.75, here correspond bounded chaoic oscillaions of he planeoid as is argued by looking a he space-ime diagram in he down picure.
5 5 HBVM(8,) Gauss 5 5 5 4 6 8 HBVM(8,) Gauss 4 6 5 5 Figure.:Upper picure:relaive error H(y n ) H(y ) / H(y ) of he Hamilonian funcion evaluaed alonghe numerical soluion of he HBVM(8,) and he Gauss mehod. Down picure:relaive error M(y n ) M(y ) / M(y ) of he angular momenum evaluaed alonghe numerical soluion of he HBVM(8,) and he Gauss mehod.
6 CHAPTER. NUMERICAL TESTS.5 Trajecory of m Trajecory of m.5.5.5.5.5 3 3 4 6 8 5 5 5 3 35 Figure.:The Sinikov problem solved by he Gauss mehod of order 4, wih sepsize h =.5, in he ime inerval [, 5]. The rajecories of he primaries in he xy-plane (upper picure) exhibi a very irregular behavior which causes he planeoid o evenually escape he sysem, as illusraed by he space-ime diagram in he down picure.
7.5 Trajecory of m Trajecory of m.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5 5 5 5 3 35 Figure.3:The Sinikov problem solved by he HBVM(8,) mehod (order 4), wih sepsize h =.5, in he ime inerval [, 5]. Upper picure:he rajecories of he primaries are ellipse shape. The discreizaion inroduces a ficiious uniform roaion of he xy-plane which however does no aler he global symmery of he sysem. Down picure:he space-ime diagram of he planeoid on he z-axis displayed (for clearness) on he ime inerval [, 35] shows ha, alhough a large value of he sepsize h has been used, he overall behavior of he dynamics is well reproduced (compare wih he down picure in Figure.).
8 CHAPTER. NUMERICAL TESTS 6 5 4 3 5 5 5 3 35 6 5 4 3 5 5 5 3 35 Figure.4:Disance beween he wo primaries as a funcion of he ime, relaed o he numerical soluions generaed by he Gauss mehod (upper picure) and HBVM(8,) (down picure). The maxima correspond o he disance of apocenres. These are conserved by HBVM(8,) while he Gauss mehod inroduces pachy oscillaions ha desroy he overall symmery of he sysem.