Iulian Coroian

Directory of Open Access Journals

REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATIONTorne 29, No 1, 2000, pp. 15-21ON A CLASS OF DIMSIMs METHODSIULIAN COROIANAbstract. We provide conditions concerning the partitioned block matrix from theRunge-Kutta methods in order to obtain methods of given order.1. INTRODUCTIONThe general linear methods for the numerical solutions of the initial valueproblem(1.1) v r a )(1.2) a(ro) :,uowithgiven f ,lo,b]xR ) R^,lJo: a, Ao € R*, introducedbyJ. C.Butcher, [1], have been discussed by many authors, [1] [Z]. This type of nu-merical methods includes, as particular cases, many known methods: linearmultistep methods, Runge-Kutta methods, predictor-corrector methods, etc. "The general linear methods with s stages are defined by:s r(1.J) yl"+r] : hDa¿jÍ(rn t c¡h, y;"*t\ +D,"0¡y?); i: r,2,j:t j:t(1.4) al**'t: h t b¿¡ I @n t c¡h, y;"*'\ +f ,oi{t; ,i : r,2, ...,r,j:t j=rwhere n :0, I,2, ..., N - 1, h: (b - o)lN, tn: r0 I nh, fl:0,I,2,...,N.The methods given by (1.3)-(1.4) can be represented as a partitioned(s + r) x (s * r) matrix2000 AMS subject classification: 65L05, 65L07, 6óL20.16 Iulian Coroian(1 .5)withA: (o¿¡); B: (b¿¡),U : (u¿j),V : (u¿¡).If the matrix ,4. is strictly lower triangular, then the general linear meth-ods (1.3)-(1.4) are called explicit and in opposite case, implicit. The diagonallyimplicit multistage integration methods (DIMSiMs), investigated by Butcherand Jackiewicz, l2], [3], have a matrix ,4 lower triangular, with a constantvalue on the diagonal. This class of methods can be divided into four types,[3], depending on the structure of the matrix ,4..The vector. y.['*t], i :T], called internal stages of the method, giveapproximations of order q (the stage order) for the solution of the problem(1.1)-(1.2) at points rn + cih, i, : lJ, U["+t) cailed internal stages, give ap-proximations of order p (the order of method), of the solution of the problem(1.1)-(1.2) at the point z,r-,.1 : rnl h. We recall that the general linear meth-ods (1.3)-(1.4) have the stage order q and the order p, if there exist vectorsak: (o4k, (r2tt, ..., an*)T, lc :0,7r 2, ..., p, such that under the assumptionp2. OR,DER CONDITIONS FOR A NEW CLASS OF DIMSIMsFirst, we recall one result of Butcher, from [B], expressed byTnpoRpn' 2'r' (Theorem s.1 from [sJ) The method, (1 s)-(1.Ð has or-rler p - q, i,f the matrices A, B, (J,'V sailify \(2.1) e'" : zAe." I Uw + Oen+\,(2'2) ez . ,u) : zBe., -f Vw + Oen+\,wherec:(cttc2r...rcr)Tre.":(ectzr€"rrr..,re""r)T and,w:(wtrw2r..,)ur)is the uector wi,th componentspwi : Daikzk, i : I, 2, ..., r.h:oNow, we will extend the conclusion.of the theorem 5.1, [3], varid onry ifthe matrix [/ is the identity matrix, by the next theoremTupoRpr¡ 2.2. The DIMSIMtJ22DIMSIMs Methods 77(Ê Y,)(2.3)has the order p: q: r : s, i,f and, only if, the mo,tri,ces A, B, U, V satisfy(2.4) UV : Vuand(2.5) UB : Bo - AB1 - VBz +VA,uhere Bo, Bt, 82 are matrices of ord,er s hauing respectiuely the erementsABUV )(1.6) a:"1 :\a¿¡y&) @ò n* + o (he+r) ,fr:0one has(1.7) Yol"l : y@n + c¿h) 1- o(hø+t¡, i : 1,2,p,s, (2.6) t9,: ü*"t e¡@)d'r , j' )"oi r, r.r"'l- * ffi ,' ̂ '),e¡ (r) :(1 .B) ,ln+rl - \,a¡y(*) (rn + h) + Ogf+r¡k:0The matrix V determines the stability of the method (1.3)-(1.4). Themethod is stable (or zero stable) if V is power bounded, i. e.(1.9) sup lllz"ll < oo.The aim of this paper is to highlight a class of DIMSIMs of the type 1,with p : q: r : s, in section 2,by an extension of a result of J. C. Butcher([3], Theorem 5.1). Also we derive a family of such stable methods dependingon two parameters, in section 3.Proof. Using Theorem 2.I and (2.4), we obtain(2.7) (Jw : (I - zA)e." + Oen+\,(2.8) e"[Jw : z(JBe"" -f VUw I Oen+rS,with 1- the identity matrix.FÞom (2.2) and (2.8) we obtainVUw : Ve'" - V Aze"" + OQv+I),(1+ c¿)ejkj) )sil @-"t)k:I, kliIulian Coroian 4 o DIMSIMs Methorls 19i8(2.e) (U B - V A)2e"" : e'' e"" - A(ze"e"") - Ve"" 1- OQn+r¡The proof follows immediately. In this case, the order condition (2.b) canbe written as the system.,btt * 1bzt: J:1 t2li +ù + t'þún + (o + 13 - þt)bzt : ?? -', .(3.4) I- t-LL ' \-" ' r '-"'o' 2(I+l)'abn I þbzz: ==1-,2(1 + ì'þtbn+(o+þ-þùbzz-2- * z(r+ù'From the system (3.4) followsFor z:0, we see that the conclition Ve: e: (1' 1'1b finish the proof, with (2'9) we follow exactly theand obtain the conclusion (2'5)'..., 1)T € -R', hold'Butcher's Proof, [3],!Remarlç 2.1. The order condition (2.5) from Theor9m2.2, offers the pos-sibility to construct a class of DiMSIMs of ãrder p, which includes the generalclass found bY Butcher, [3], P' 356'3. A FAMILY OF STABLE DIMSIMs OF ORDER 2Inihefollowing,usingTheorem2'2',wewilld'eriveafamilyofstableDIMSIMs wittr p : n -- '": ':2 of type 1' depending on two parameters'f'o, this, we select thã matrices A, B, U, V of the formbtt :br2:bnbz2 :(2cr-rt)@-þù+p(I+j)2(a-r p)(r + t)@ - þt) 't@-þt)+(2a-3)BQ+1)(3.5) 2(a + þ)(t +.y)(" - þt)(2a-I)a-fu(2a+1)t:(0\û, 3),B: (bir bnbzt bzz ))2(a+ p)Q+t)@- þt)'.tkr - þt) - (2a - 3)a(1 + 1)(3.1)2(a+ p)(1+t)@- þt)U- utt u1.2u2r u22 )v- ( u I-uu I-uwith a, B, 1 free parameters, subject to a I -þ, ^f + -1, a * h.The remaining parameters a, B, 7 w1lfl, be obtained from the conditionsatisfying (2.4), (2.5).We take c: (cr,"r)' : (0,1)T and then the matrices Be' 81' B2 fuom(2.5) are(3.6) det M(z) :0.Here, the stability matrix M(r), is given bytlt_(r.z¡ Bo: [ 2\0 i)'"': ( j, 'r)' u': ( i ;) (3.7) M(r):V *zB(I - zA)-rU,with .4, B from (3.1) and t/, lz from (3.3).After a tedious computation one hasRemark 3.1. In order to apply Theorem 2'2we need matrices [/' V suchthatUV:VU (3.s)wheredetM a) : Nê+ 1), + +(å * #) u,If we take .'t 1U: (Dapþt a+ þ - 0tv 1 t'y 1 -tl N :2(a+ Þ) (1 +r) @_ þt),¡/1A : ; (1 - 2a- 3l), n : rNa(2a - 3),¡¡zC : --------- (0 - a)1 - a(2a - J))2(1 *7)..(3.3) 1^YI*l 7*l (3.e)with o, þ, I I -1 real parameters, then (2'4) hold' and for o:0' P -- 0follows that (J :,I, where -I is identity matrix'20 Iulian Coroian 67 DIMSIMs Methods 2tBy substituting (3'9) in (3'8) we obtain(3.10) detM Ø : i+ r) {(t - (zo- ¡)r) z i (2 - a)t' "}'\Me see, that for a:2 and 7: 1 followsdet M (z) :0'So, for o' : 2, .Y : l, with (3'5) we have obtained one family of DIMSIMs oftyie 1 and oi order p : 2 : q depending on two parameters. This family ischaracterized bY the matrices[5] COROIAN, I., BÀ.RBOSU, D., on the general linear method,s for initial ualue problems,Technical University Kosice, Slovakia, Matematica a jej aplikacie v technickych vedach,pp.20 24,7997.[6] JACKIEwrcz,z', VERMIGLIO,R., General rinear method,s w,ith enternal stages ol dif-ferent ord,er, BIT, 96, 4, pp. 688-71 2, 1996.[7] JACKIEwrcz, z', VERMIGLIO, R., ZENNARO, M., variabre step size d,iagonalry ,im-plicit multistage 'integrat,ion method,s for ord,inary d,ifferentiat eqiation, Appl. Numer.Math., 16, pp. 343-367, 199b.Received September 10, lgggNorth Un,iuersity of Baia MareD epartment of M athemat,icsand, Computer ScienceVictoriei 76,/800 Baia Mare RomaniaE-mail: icoroian@univer. ubm, ro5a-38 1o: (0, 4(az - Uz,3a-5Pa(L; þ)(3. 11) 4(o2r12Ll2_p2112r12+(" - þ)U- o¿pwhere a, þ € R, a * þ, o I -8, are free parameters'Remarlc 3'2' Because ll7"ll : !, n € N*' the matrices (3'11) providestable methods.Remarlc3.3'Ifweselectforparametersa,Bthevaluesa:1'É:0weobtain the DIMSIM given by the matrices,:(9 3u: (å ÎB514314r12L12rl4-rl41/2 \u2)(3.12),V:This particular DIMSIM has been found by J. C. Butcher, [3], section 6, usingas matrix U line identitY matrix 1REFERENCES[1] BUTCHER, J. C.' Th,e Numerical Analysis of Ord'i'nary Differential Equations' RungeKutta anil General L,inear Methoils, John wiley and sons, chichester and New York,1987.[2] BUi6HER, J. c., JACKIEWICZ, 2., D'iagonally irnpli,cit general linear method's forord,inarg d,'ifferential equations, BIT, 33, pp' 452-472,1993[3] BUTCHE¡, j. C., Diagonallg Impli,c,it Multi-Stage Integration Methoils, Appl. Numer.Math, 11, PP. 374-363' 1993'[4] COROIÁN, i., O, the general linear integration methods, Bul. $t. Univ' Baia Mare, SeriaB, Matematicä-Informaticä,, 13, pp' 5l-57, 1997'

On a class of DIMSIMs methods

https://www.ictp.acad.ro/jnaat/journal/article/download/2000-vol29-no1-art2/Coroian

On a class of DIMSIMs methods

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals