A Functonally Ftted 3-stage ESDIRK Method Kazufum Ozawa Akta Prefectural Unversty Hono Akta 05-0055, Japan ozawa@akta-pu.ac.p Abstract A specal class of Runge-Kutta (-Nyström) methods called functonally ftted Runge-Kutta (FRK) methods has recently been proposed by the author. Ths class of methods s based on the exact ntegraton of a gven set of functons, whch s called the bass functons by the author, and as a result, the method s always exact, f the soluton of the ODE s expressed by a lnear combnaton of the bass functons. In ths paper, a 3-stage explct sngly dagonally mplct Runge-Kutta (ESDIRK) method n ths class s developed. It s shown that the global error of the proposed method behaves lke O(h 4 ), where h s the step-sze, even for the cases that the method s not ftted completely to the equaton. The method s extended to an embedded par. Several numercal experments show that the method ntegrates the ODE very accurately, for the case that a sutable set of the bass functons can be found, and produces reasonable accuracy even for general cases. Introducton Intal value problems of ODEs are very mportant tools n scence and technology. When solvng the problems, t s often the case, n practce, that a pror nformaton on the soluton and/or equaton, such as the perod of the soluton or the domnant egenvalue of the coeffcent matrx, s avalable. For ths case, f we could desgn a numercal method based on such nformaton, then the method would be very accurate for ths problem. For example, f the soluton of the ODE s a snusodal functon wth a small perturbaton, then the specal method whch s exact only for the trgonometrc functon wth the frequency wll be more accurate than general ODE methods. Many specal methods whch are exact for trgonometrc functons, exponental functons, or mxed-polynomals have been derved (see e.g. [], [7], [0], [], [3], [4]). As an extenson of these method, Ozawa has recently developed the Runge-Kutta (-Nyström) method that s exact on the lnear space of any gven functons ([8], [9]). The method s also an extenson of the collocaton Runge-Kutta methods, whch are the specal cases that the functons are polynomals. The method proposed by Ozawa, lke collocaton methods, s fully mplct, so that ts computatonal cost s extremely expensve compared wth explct methods, and expensve wth dagonally mplct Runge-Kutta (DIRK) methods. The purpose of ths work s to develop a computatonally cheap Runge-Kutta method whch are exact for the gven functons, usng the smlar technque used n Ozawa ([8] and [9]). Functonally ftted Runge-Kutta method Consder the ntal value problem dy(t) dt = f(y(t)), y(0) = y 0, t [0, T ], ()
and the s-stage Runge-Kutta method y n+ = y n + h Y = y n + h s b f(y ), = s a, f(y ), =,..., s, = () where h s a step-sze, and y n s the numercal approxmaton to y(nh). Almost all Runge-Kutta methods are desgned to ntegrate the equaton exactly, f the soluton of () s any polynomal of some degree or less. In our approach, however, the Runge-Kutta method s desgned to be exact not for polynomals but for the lnear space of gven functons {Φ m (t)} s m=. We call the functons {Φ m (t)} s m= the bass functons, and the resultng Runge-Kutta method and the functonally ftted Runge-Kutta (FRK) method. One way to get the coeffcents of the FRK s to gve the functons {Φ m (t)} s m=, and then solve the (s + ) smultaneous equatons s Φ m (t + c h) = Φ m (t) + h a, ϕ m (t + c h), m F, =,..., s, = (3) s Φ m (t + h) = Φ m (t) + h b ϕ m (t + c h), m F s+, = for the unknowns a, and b, where ϕ m (t) = Φ m (t), and F F {,,..., s} s the set of the subscrpts m of the functons Φ m (t) used n the th equaton. Here we assume that c s are constant and dfferent from each other. In [8] and [9], F s always F for each, that s, all the functons Φ m (t) (m =,..., s) are used to determne all the unknowns a, ( =,..., s), whch means the resultng method s necessarly a fully mplct one. In the present case, however, we frst gve s F coeffcents n accordance wth the sparsty pattern of a predetermned Butcher array, and then determne the remanng F coeffcents usng F dfferent functons of Φ (t),..., Φ s (t). It s clear from ths constructon that the method s always exact, f the soluton y(t) s an element of the space spanned by the Φ ν (t) s, where ν s+ = F. In Ozawa [8], the coeffcents of FRK gven by (3) are shown to be unquely determned for all h and t [0, T ], f the Wronskan matrx W (t) ϕ (t) ϕ s (t) (t) ϕ() s (t).. (t) ϕ s (s ) (t) ϕ () ϕ (s ) s nonsngular. Moreover, n [8], these coeffcents are shown to be analytc, f all of the functons {Φ m (t)} s m= are analytc n [0, T ]. In general, the coeffcents a, and b determned n ths way depend not only on h, but also on t. We shall consder, however, the case that these coeffcents depend only on h; f the bass functons Φ m (t) are polynomals, exponentals or snusodal functons, then ths s the case. By ths assumpton, we set t = 0 n (3) wthout loss of generalty. (4)
3 Local truncaton error of FRK method The numercal results gven by FRK wll have truncaton errors, except for the case that FRK s ftted to the problem () completely. Therefore, t s necessary to evaluate the error by some measure. In ths paper, we use the order of accuracy to evaluate the error, as n the case of conventonal numercal methods of ODEs. The defnton of the order of accuracy of FRK s the same. That s, f the numercal soluton by FRK satsfes y y(h) = O(h p+ ), y(0) = y 0, h 0, for any suffcently smooth functon y(t), then we shall call the nteger p the order of accuracy of FRK. However, unlke the conventonal case of constant coeffcent methods, we must consder the errors n the stuaton that the coeffcents a, and b also vary as functons of h, when h 0. In order to analyze the local truncaton error of FRK, let us ntroduce the followng quanttes: B(q) C (q) D(q) b c q q, (5) a, c q cq q, (6) b C (q). (7) Accordng to Ozawa [8], [9], these quanttes satsfy B(q) = O(h r s++ q ), q =,..., r s+, C (q) = O(h r + q ), q =,..., r, where we set r = F ( =,,..., s + ). We express the errors at the stages and step by usng B(q) and C (q). Frst we consder the resduals at the stages and step. Let y(t) be any suffcently smooth functon (not necessary the soluton of ()), then R y(0) + h R y(0) + h b y (c h) y(h) = q a, y (c h) y(c h) = q h q B(q) (q )! (y (0)) (q ), Note that y(t) = Φ m (t) these resduals vansh, whch mples h q B(q) (q )! (ϕ m(0)) (q ) = 0, m F s+ q q h q C (q) (q )! (ϕ m(0)) (q ) = 0, m F. h q C (q) (q )! (y (0)) (q ). On the other hand, f Φ m (t) are polynomals of some degree or less, then B(q) and C (q) vansh for the frst several q s, and ϕ (q ) m (t) = 0 for the other q s. From (8) and (9) we have R = O(h r+ ), R = O(h ρ+ ), () (8) (9) (0)
where ρ = mn{r }, r = r s+. Next we consder the relaton between the resduals and local errors of FRK method. Let y(t) be the soluton of y (t) = f(y(t)), then the errors at the stages are gven by therefore e Y y(c h) ( = y 0 + h a, f(y ) y 0 + h = h f y a, (e + O(e )) + R, e = ( a, h f y ) ( (h f y ) a, y (c h) R ) a, (e + O(e )) + R ) = O(h ρ+ ). () For the error at the step, we have E y y(h) = y 0 + h b f(y ) ( y 0 + h = h b (f(y ) f(y(c h))) + R = h f y b (Y y(c h) + O(e )) + R. b y (c h) R Before evaluatng E, we must evaluate b Y = b y 0 + h b a, f(y ),, b y(c h) = b y 0 + h b a, y (c h) T,, where we put T = For the order of T, f we assume then from () we have Thus E = (h f y ), = (h f y ), b R = q ) (3) h q D(q) (q )! (y (0)) (q ). (4) T = O(h τ+ ), (5) τ ρ = mn {r }. b a, (f(y ) y (c h)) + (h f y ) T + R + O(h ρ+3 ) b a, e + (h f y ) T + R + O(h ρ+3 ).
If the order of, b a, e s that of the mnmum of e s, then the order of accuracy p of the method s gven by p = mn {ρ +, τ +, r}. (6) 4 3-stage FESDIRK method Let us consder the 3-stage Runge-Kutta method gven by the Butcher array 0 0 c a, α c 3 a 3, a 3, α b b b 3 (7) Usually the methods of ths type are called explct SDIRK (ESDIRK) method when the coeffcents are constant, and we shall call t functonally ftted ESDIRK (FESDIRK) method, f the method s FRK. For the gven c s and the gven {Φ m (t)} 3 m=, we determne the coeffcents of (7) by Φ m (c h) = Φ m (0) + h (a, ϕ m (0) + α ϕ m (c h)), m =,, Φ m (c 3 h) = Φ m (0) + h (a 3, ϕ m (0) + a 3, ϕ m (c h) + α ϕ m (c 3 h)), m =,, Φ m (h) = Φ m (0) + h (b ϕ m (0) + b ϕ m (c h) + b 3 ϕ m (c 3 h)), m =,, 3, (8) where we assume that the Wronskan matrx ϕ (t) ϕ (t) ϕ 3 (t) W (t) = ϕ (t) ϕ() (t) ϕ() 3 (t) (9) ϕ () (t) ϕ () (t) ϕ () 3 (t) s nonsngular at t = 0. Ths method s exact for y(t) span{φ (t), Φ (t)}. For ths case, we have r = r 3 =, r 4 = 3, ρ =, τ, and B(q) = C (q) = 3 = 3 = b c q q = O(h4 q ), q =,, 3, a, c q cq q = O(h3 q ), q =,, whch leads to p = 3 from (6). We shall call the method FESDIRK3. When h 0, FESDIRK approaches a constant coeffcent method, whch has a key role n later consderatons. Here we consder the coeffcents of ths constant coeffcent method. Relaton (0) means that 3 = = (0) c q =, q =,, 3, () q a (0), cq = cq, q =,, () q
where a (0), and are the constant terms n the power seres expanson n h of the coeffcents. The relatons () and (), whch are the so-called smplfyng assumptons [], determne a (0), and b(0) unquely as functons of c. The results are: Note that a (0), and b(0) a (0), = c, a(0), = c (= α), a (0) 3, = 36 c4 0 c3 + 34 c 60 c + 9 8 c (3 c ), a (0) 3, = 4 c3 50 c + 36 c 9 8 c (3 c ), a (0) 3, 3 = α, = 6 c 6 c + 6 c (4 c 3), = 3 = 6 c (6 c 8 c + 3), (3 c ) 3 (4 c 3) (6 c 8 c + 3). gven above are ndependent of the choce of Φ m (t). 5 Fourth order FESDIRK method We have obtaned a 3-stage FESDIRK method, whch we call FESDIRK3, and have shown that the method s of order 3. In order to rase the order of the method up to 4 we assume two condtons. The frst condton s 0 t q t (t c ) (t c 3 ) dt { = 0, q =, 0, q. We wll consder later the case that the ntegral equals to 0 also for q. From ths assumpton we have c 3 = 4 c 3 (3 c ). (4) By assumng (4), we have from [8] B(q) = 3 = so that r = 4 n (), and we have, nstead of (), (3) b c q q = O(hmax{5 q, } ), q =,..., 4, (5) 3 = c q =, q =,..., 4. (6) q The second assumpton s a (0), = b(0) ( c ), =,, 3. (7)
It has been shown that ths condton together wth () and (6) s a suffcent condton for the method (a (0),, b(0), c ) to be of order 4 (see [], [4]). Next lemma proves that condtons (), (6) and (7) guarantee τ = 3 n (5). Lemma If condtons (), (6) and (7) hold, then D(q) = O(h 4 q ), q =,, 3, so that τ = 3 n (5). Proof. Let the power seres expanson of D(q) be D(q) = D (0) (q) + D () (q) h + D () (q) h + From the defnton of D(q) n (7) and the property of C (q) gven by (0), we have mmedately D() = O(h ), D() = O(h), or equvalently D (0) () = D () () = 0, D (0) () = 0. Next we show that several terms other than the above vansh. From (), (6) and (7), we have for q =,, 3 D (0) (q) = ( ) a (0), cq cq = ( c ) c q = 0. (8) q q (q + ) On the other hand, from (4) and (8) q h q D(q) (q )! (ϕ m(0)) (q ) = ν ( ν q= = 0, m =,. (ϕ m (0)) (q ) (q )! D (ν q) (q) ) h ν (9) Therefore, the condton that the coeffcent of h 3 n (9) s beng 0 can be expressed by ϕ (0) D () () + ϕ () D () () + ϕ() D (0) (3) = 0, ϕ (0) D () () + ϕ () D () () + ϕ() D (0) (3) = 0. Snce we have already had D (0) (3) = 0 n (8), and the submatrx ( ) ϕ ϕ ϕ () ϕ () of Wronskan matrx (9) s nonsngular by assumpton, then (30) (3) D () () = 0, D () () = 0.
Summarzng the results obtaned so far, we have D (0) () = D () () = D () () = 0, D (0) () = D () () = 0, D (0) (3) = 0. It s clear from the dscusson n ths lemma that any other terms of D (l) (q) never vansh. Thus we have proved ths lemma. Snce r = 4 has already been establshed, and τ = 3 has been proved by the above lemma, t s clear from (6) that p = 4. Thus we have the followng theorem: Theorem If the abscssae c and c 3 satsfy the two condtons (4) and (7), then FESDIRK wth the coeffcents gven by (3) s of order 4. Hereafter we call the (F)ESDIRK of order 4 (F)ESDIRK4. Next we must obtan the values of c for whch condton (7) s vald. Let d be d = a (0), b(0) ( c ), =,, 3, then from () and () we have d c q =, = q that s a (0), cq c q q + q + d + d + d 3 = 0, c d + c 3 d 3 = 0. ( c ) c q = 0, for q =,, Ths means that f we force one of d s to be 0, then the remanders become 0, provded that 0 < c c 3. Thus we put, for example, d = (3 c ) (3 c ) (c ) 6 c (4 c 3) whch leads to c = 3, 3,. Among these solutons, c = /3 s not allowed because of (4), so that we consder the remanng two solutons. Next we show the stablty regons of the ESDIRK4 s wth c = /3 and c =, and compare these regons wth that of the classcal Runge-Kutta method (RK4). = 0,
4 0 - -4-6 -4-0 Fg.. Stablty regons of ESDIRK4 s wth c = 3 (sold), c = (dashed), and RK4 (dash-and-dotted). Fg. shows that the ESDIRK4 wth c = /3 s preferable to the ESDIRK4 wth c =, snce the former has broader stablty regon. Therefore we take c = /3 also for FESDIRK4, snce t s expected that FESDIRK has approxmately the same propertes as those of ESDIRK, when h s small. Here we show the Butcher array of the ESDIRK4 wth c = /3. 0 0 3 5 6 6 4 0 Hereafter, we smply denote the ESDIRK4 and FESDIRK4 wth c = /3 by ESDIRK4 and FESDIRK4, respectvely. Fnally we nvestgate the attanable order wth the FESDIRK of the type (7). It s clear from the prevous dscusson that the FESDIRK and the ESDIRK have always the same order, snce the latter corresponds to the partcular case that Φ (t) = t, Φ (t) = t, Φ 3 (t) = t 3, and the dscusson s ndependent of the choce of the bass functons. Therefore, we wll consder the attanable order of the ESDIRK nstead of that of the FESDIRK. Any 3-stage method wth c = 0 cannot be of order 6, so that the attanable order of the ESDIRK of the type (7) wll be at most 5. If the order of the method s 5, then the condton 6 5 8 6 5 (3) c q =, q =,..., 5 (33) q must be satsfed. The set of the abscssae satsfyng the condton s gven by whch s obtaned by solvng c = 6 6 0, c 3 = 6 + 6, 0 0 t q (t c ) (t c 3 ) dt = 0, q =,.
Substtutng the c and c 3 obtaned now nto (33), and solvng ths for, we have = 9, b(0) = 6 + 6, 3 = 6 6, 36 36 and the values of a (0), s whch satsfy () wth these c s are gven a (0), = 6 6, a (0), 0 = 6 6, 0 a (0) 3, = 6 + 6 00, a(0) 3, = + 7 6, a (0) 3, 3 = 6 6. 50 0 Unfortunately, the set of the values lsted above does not satsfy some of the order condtons even for p = 4. For example, the order condton for the tallest tree of order 4, whch s gven by a (0), a(0),k a(0) k,l = 4,,,k,l s not satsfed wth these values; ths value becomes 57 6 00 for the present set of the coeffcents. Thus we have the concluson that the attanable order wth the FESDIRK s 4. Ths means that FESDIRK4 s the hghest order method n the class of (7). 6 Numercal example In order to see how well FESDIRK4 s ftted to the specal problems for whch we can fnd the bass functons, and whether or not the global error of the method behaves lke O(h 4 ) for general problems, we shall some present numercal examples. The problems to be solved are: A. Ary equaton B. Bessel problem C. Constant coeffcent lnear equaton D. Duffng equaton Problem A s the one whose soluton oscllates wth varyng frequency. Problems B and D are perturbed oscllatons, and C s the problem whose soluton conssts of the two components: rapdly damped oscllatory component and decayng exponental component. In order to generate the coeffcents of FESDIRK4, we use snusodal bases for problems A, B and D, and exponental bases for problem C. In these experments, we compare the Eucldean norm of the errors wth those of the other methods. All the computatons are performed by the IEEE double precson arthmetc. Ary equaton Consder the Ary equaton y (t) t y(t) = 0, (34)
wth the ntal condton y ( 50) = A( 50) + 0.5 B( 50) =.304564997 0, y ( 50) = A ( 50) + 0.5 B ( 50) = 3.96308987 0, where A (t) and B (t) are Ary s A and B functons, whch are lnearly ndependent solutons of Eq. (34) (see [6]). The exact soluton of the problem s For ths problem, the bass functons y(t) = A(t) + 0.5 B(t). Φ (t) = t, Φ (t) = cos(ω t), Φ 3 (t) = sn(ω t), (35) wll be approprate, n whch case the Wronskan matrx assocated wth the functons s nonsngular. We ntegrate the equaton from t = 50 to 0, changng the angular frequency ω by the formula ω = t, at every nteger pont t = 50, 49,..., 0. The error of FESDIRK4 s compared favourably wth that of ESDIRK4 n Fg., although both of the methods are 4th order accurate. 0 log E -0-0 -30-40 -50 4 6 8 0 -log h Bessel problem [5] Next, let us consder the equaton wth the ntal condton Fg.. Errors E of FESDIRK4 (sold) and ESDIRK4 (dashed) versus step-sze h for Ary equaton (34). y (0.5) = 0.5 J 0 (5) y (t) + 00 y(t) = y(t) 4 t, (36) =.5579883 0 y (0.5) = J 0(5) 0.5 0 0.5 J (5) =.9075444 The exact soluton of the problem s gven by y (t) = t J 0 (0 t),
where J ν (t) and Y ν (t) are the Bessel functons of the frst and second knd, respectvely. We ntegrate the equaton from t = 0.5 to 0 by the two methods: FESDIRK4 wth ω = 0 (fxed) and ESDIRK4. The results are shown n Fg.. From the result, we can observe that the two methods are of order 4 also n ths example, and that the accuracy of FESDIRK4 s remarkable compared wth the prevous example. 0 log E -0-0 -30-40 -50 4 6 8 0 -log h Fg.. Errors E of FESDIRK4 (sold) and ESDIRK4 (dashed) versus step-sze h for Bessel problem (36). Constant coeffcent lnear equaton The thrd problem to be consdered s the lnear homogeneous equaton where y (t) = P y(t), y(0) = (, 0, 0, 0) T, (37) 0 0 0 P = 96 97 6 98 0 99 96. 0 0 The exact soluton of the problem s gven by e t + e 00 t sn t e y(t) = t ( + t) + e 00 t (cos t + sn t) e t + e 00 t (cos t + sn t). e 00 t sn t Ths soluton conssts of fast and slow modes. If the step-sze n the stablty regon s used, then the fast mode wll be damped out very soon and, as a result, the slow mode wll domnate the entre soluton. Hence, we ft the method to the slow mode, that s, we choose the followng bass functons: Φ (t) = t, Φ (t) = exp( t), Φ 3 (t) = t exp( t). (38) We ntegrate the equaton from t = 0 to by the FESDIRK4 wth bass functons (38), and compare the error wth those of the three Runge-Kutta methods: ESDIRK4, the
-stage Gauss (Gauss) and the classcal Runge-Kutta (RK4) methods, each of whch s of order 4. The results are shown n Table. Table. Errors of varous methods for lnear equaton (37). log h log E FESDIRK4 ESDIRK4 Gauss RK4.708e+0.95e+0-5.4e+00.099e+0 3.486e+0.73e+0 -.96e+0.53e+0 4 -.858e+0 -.585e+0 -.59e+0.68e+0 5-5.334e+0 -.985e+0 -.99e+0 4.70e+0 6-5.7e+0-3.387e+0-3.39e+0-3.068e+0 7-5.6e+0-3.787e+0-3.79e+0-3.470e+0 8-5.5e+0-4.88e+0-4.9e+0-3.870e+0 9-5.09e+0-4.586e+0-4.530e+0-4.70e+0 0-5.64e+0-5.073e+0-4.907e+0-4.668e+0-5.e+0-5.056e+0-5.3e+0-5.63e+0-5.06e+0-5.06e+0-4.988e+0-5.078e+0 E s the Eucldean norm of the error at t =. It can been seen that, although FESDIRK4 s less stable than the -stage Gauss Runge- Kutta method for large step-szes, ths method s ftted to the problem completely for moderately small step-szes; the values of order 50 or less n the second column of the table must be the accumulatons of round-off errors, snce the machne epslon of the IEEE double precson arthmetc s 53. On the other hand, the errors of the other methods decrease very slowly at the rate of O(h 4 ). Duffng equaton [3] The last one to be ntegrated by FESDIRK4 s a nonlnear equaton. Let us consder the Duffng equaton y (t) + µ y(t) = k (y(t) y(t) 3 ), (39) The exact soluton s gven by y(0) = 0, y (0) = µ. y(t) = sn (µ t; (k/µ) ), where sn( ; ) s the Jacoban ellptc functon. We ntegrate the equaton wth µ = and k = 0.03 from t = 0 to 00 by the FESDIRK4 wth the bass functons (35)(we set ω = ) and ESDIRK4. The result s shown n Fg. 3. For ths example, both of the methods are more accurate compared wth Problems A and B, n spte of the long term nterval of ntegraton. To summarze, FESDIRK4 s a very effcent scheme for the specal problems for whch we can fnd the bass functons successfully, and s reasonably accurate for general problems, unless the problem s very stff. Ths s due to the fact that the method has 4th order accuracy for general problems.
-0 log E -0-30 -40-50 4 6 8 0 -log h Fg. 3. Errors E of FESDIRK4 (sold) and ESDIRK4 (dashed) versus step-sze h for Duffng equaton (39). 7 Embedded FRK method Snce we have solved Problems A, B, C, and D, next we must consder E (embedded) method. Let us consder the embedded par y n+ = y n + h (b f(y ) + b f(y ) + b 3 f(y 3 )), ȳ n+ = y n + h ( b f(y ) + b f(y ) + b 3 f(y 3 ) + b 4 f(y 4 )), (40) where Y = y n, Y = y n + h (a, f(y ) + α f(y )), Y 3 = y n + h (a 3, f(y ) + a 3, f(y ) + α f(y 3 )), Y 4 = y n + h (a 4, f(y ) + a 4, f(y ) + a 4, 3 f(y 3 ) + α f(y 4 )), and we assume c = /3 and c 3 =, as before. The Butcher array of the par s 0 0 3 a, α 5 6 a 3, a 3, α a 4, a 4, a 4,3 α b b b 3 0 (4) b b b3 b4 In the array, we further assume b = a 4,, b = a 4,, b3 = a 4, 3, b4 = α, so that the method to calculate ȳ n+ becomes FSAL (frst same as last). The computatonal cost of ths method s approxmately the same as that of FESDIRK4, snce the number of LU decomposton to be performed n the step s stll one. Here we must determne the coeffcents of the method. We take the same a,, α and b as those of FESDIRK4, so that the order p of the method corresponds to b s 4. Wth
these coeffcents, f we choose the b such that the order of the method corresponds to b s 3. If we force then we have R y(0) + h B(q) 4 = b c q q = O(h4 q ), q =,, 3, (4) 4 b y (c h) y(h) = q = Therefore, f we set R = O(h r+ ), then r = 3 and h q B(q) (q )! (y (0)) (q ) = O(h 4 ). p mn {ρ +, τ +, r} = mn {4, 4, 3} = 3. (43) The coeffcents b, b and b 3 satsfyng (4) are gven by solvng the system of the equatons Φ m (h) = Φ m (0) + h ( b m ϕ m (0) + b ϕ m (c h) + b 3 ϕ m (c 3 h) + α ϕ m (h)), m =,, 3, for the gven {Φ m (t)} 3 m=. Thus, we have the 3rd order method embedded n the 4th order one. The step-sze strategy for ths par, whch controls the local truncaton error of the lower order method wthn a prescrbed tolerance T OL, s gven by ( ) /4 T OL h n+ = θ h n, ȳ n y n where θ s a safety factor, say θ = 0.9. Now, let us apply the embedded par to the two-body problem [5], [] wth the ntal condton y = y /r 3, y = y /r 3, r = y (0) = e, y (0) = 0, y (0) = 0, y (0) = + e e, y + y, (44) where e (0 e < ) s an eccentrcty. The exact soluton of ths problem s where u s the soluton of Kepler s equaton y (t) = cos u e, y (t) = e sn u, u = t + e sn u. The soluton of (44) s found to be π-perodc for any e. Hence, a natural choce of the bass functons s (35) wth ω =. By ths choce, the problem wth small e s expected to be accurately solved, snce the soluton s purely snusodal when e = 0. We ntegrate the problem wth e = 0.005 from t = 0 to 50 π by the two embedded pars FESDIRK4(3) and the ESDIRK4(3). From the result of Table, we can see that the embedded method derved here controls the local truncaton error well, and as a result, the method ntegrates the equaton wth fewer steps compared wth ESDIRK4(3).
Table. Errors and the total steps for two-body problem (44) wth e = 0.005. FESDIRK4(3) ESDIRK4(3) T OL error steps error steps 0.785e+00 5.483e+00 36 0 3.866e-0 70.53e+00 77 0 4 7.846e-03 5.494e-0 496 0 5.399e-03 38 9.359e-03 884 0 6.690e-04 680 6.00e-04 573 0 7.846e-05 07 4.46e-05 796 0 8.938e-06 44 3.4e-06 4970 0 9.993e-07 3806.848e-07 8833 0 0.0e-08 676.530e-08 5706 T OL: Tolerance of the local error. error: Eucldean norms of the errors at t = 50 π. steps: Total number of tme steps (ncludng reected steps). 0.44 h 0.4 0.4 0.38 0.36 Error 0.00 0.00 0.0008 0.0006 0.34 0.0004 0.3 0.000 0.3 0 0 40 60 80 00 0 40 60 t 0 0 0 40 60 80 00 0 40 60 t Fg. 5. Step-sze plot and error behavor of embedded par FESDIRK4(3). 8 Summary and future work We have presented a functonally ftted 3-stage ESDIRK method. Although the method s of order 4 for general cases, the method s always exact when the soluton of the ODE can be expressed by a lnear combnaton of the gven bass functons. Varous numercal examples show that the method has proved successful when a sutable set of the bass functons s found, and reasonably accurate, even f ths s not the case. The method s extended to an embedded par. The stablty analyss and the mplementaton ssue of FRK wll be future works. Acknowledgment. The author would greatly apprecate the conference organzers, Prof. S. Maeda, Prof. T. Ito and Prof. T. Mtsu, for ther efforts. The author also wshes to thank Mr. C. Hrota, the research assstant of Akta Prefectural Unversty. He provded much support for ths work.
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