SET OF METHODS FO SOUTION THE AUHY POBEM FO STIFF SYSTEMS OF ODINAY DIFFEENTIA EUATIONS AF atypov and YuV Nulchev Insttute of Theoretcal and Appled Mechancs SB AS 639 Novosbrs ussa Introducton A constructon ethod of ult-curve paraetrcal polynoals of an arbtrary power s descrbed; and evaluaton of nterpolaton accuracy and quadrature forulas are presented Usng the nterpolatons descrbed two new sets of soluton ethods of the auchy proble for the syste of ordnary dfferental equatons (ODE) were developed angers of paraeters where A-and ethods are stable are deonstrated Mult-curve polynoals et the functon f(x) at the segent [x x K ] be gven at K values of the arguent < x < x K wth ts dervatves up to the orders K x < Denote the functon dervatves f(x) n gven ponts f () f( x ) ( ) nclusvely ( ) f so that f d f( x )/ dx Denote also X x x ] + Assung that the functon f(x) s sooth up to the power M [ + + ax( + ) + K wthn the whole segent [x x K ] we wrte t at the segents X K as polynoals of the followng for: + + ( ) ( ξ) ξ + ξ ( ξ) f x a b ξ ( x x )/ h h x x + A chosen for allows one to wrte the relatonshps for deternng the polynoal ( ) ( ) coeffcents () by the gven values f f+ We obtan () ( ) ( ) + + + a f /! b f ( ) /! ( ) ( )! f h f K! ( )! () Usng relatonshps () polynoals () are wrtten n the for where ( ) ( ) + f ( x) f ψ ( ξ) + ( ) f χ ( ξ) ξ [] (3) AF atypov and YuV Nulchev 53
ψ ξ ξ ξ + + ( ) p! + + χ( ξ) ( ξ) q ( ξ) (4)! p ( ) q ( ) ξ [] + + + + + + + + + + The forulas for defnte ntegrals of polynoals (4) have the for ξ + + + ξ ξ I ( ξ) ψ ( ξ) dξ p ( + )!! ( + + ) ξ + + + ( ξ) ( ξ) J ( ξ) χ ( ξ) dξ q ( + )!! ( + + ) (5) The followng sple expressons are gven for the values at the rght end of nterval (5): I () + ( + )! J () + + ( + )! + + + + + It s obvous that the ult-curve polynoal represented by K polynoals (3) s sooth wthn the whole segent [x x K ]uptothepower n( ) K Snce polynoal (3) at the -th nterval s an nterpolatng Hert polynoal by two ponts the resdual ter for x (x x + ) accordng to [] s wrtten by the relaton ( ) + + f ( η) h f( x) f ( x) ( ξ ) ξ ξ () η ()! (6) fro whch consderng + + + ( + ) ax ( ξ ) ξ ξ ( ) ( + ) + one obtans the upper estate of nterpolaton accuracy by polynoal (3) δ d ax x ( x x+ ) ax x ( x x+ ) f ( x ) f ( x ) f ( ) ( x ) ( + ) + ( + )! + h d (7) uadrature forulas et the calculaton ethod of a defnte ntegral for the functon f(x) be an exaple of applcaton of presentaton (3) b S f ( x ) dx a 54
et the functon f(x) be presented at the secton [a b] by polynoals (3) and (4) Then ntegral S for a gven nuber K of segent dvson s wrtten by the relaton K ( ) ( ) S h I() f + ( ) J() f+ f f( a) f f( a+ h ) > (8) For unfor dvson of the segent and a constant power of the nterpolatng polynoals forulas (8) have the for: ( ) ( ) h I () { f ( a) + ( ) f ( b) } K S K ( ) ( ) ( ) h I () f ( a) + ( ) f ( b) + + ( ) f K > f f( a+ h( )) K + enter(( b a) / The estaton of accuracy of forula (7) follows fro (5): x + [ f x + d h ( x ) f ( x )] dx ( + ) ( + )! ( + )! (9) auchy proble for systes of ordnary dfferental equatons et us apply the nterpolatons presented to the ntal value proble for a syste of n ordnary dfferental equatons: y ( x ) f ( y) y() y x [ X ] () onsder the soluton of the syste at one step h x [ x x + h ] (further we denote hh ) et the syste at the step h be represented as follows: dy / dξ h f ( y) Ф( ξ ) y() y ξ [ ] ) et us set the nteger quanttes and Assung that there s the only soluton of equatons () and the rght-part functons have a degree of soothness on y hgher than ++ we respect these functons at the step h by a su of the followng three polynoals: + + ( ξ) ( ξ) ξ ξ ( ξ) Ф a + b + + + + ( ξ) ξ ( ξ ξ ) c/ () 55
-ethods Assue n () that c / and the coeffcents of the frst two polynoals are deterned by the gven values of the functon Ф(ξ ) and ts dervatves at the boundares of the segent [ h] n the ponts ξ and ξ by relatonshps () Integratng polynoals () at the segent [ ] one obtans the followng plct syste of n algebrac equatons for the solutons at the pont ξ : ( ) ( ) y () y + I () Ф + ( ) J () Ф (3) Method of solvng of the auchy proble for ODE systes based on step-to-step soluton of the algebrac equatons (3) are called the -ethods [] The accuracy of the -ethod accordng to (9) at the ntegraton step h satsfes the evaluaton Usng (3) n the equaton [3] y( y( h z λ z z() z x e( λ) < we obtan an equaton concdng wth Pade-forulas of approxaton for the functon exp(λ [4] consequently accordng to the theore on А- stablty of Pade-approxaton [5] -ethods are A stable at the paraeters satsfyng the nequaltes + Itsalso shown that the ethod s -stable at > For applcaton n practce the splest A-stable -ethod of the 5-th order wth the paraeters can be recoended whch was successfully used to solve soe stff probles In ths case syste (3) has the for + () () y () y + ( Ф + Ф) + ( Ф + Ф ) (4) M-ethods onsder the ethods of representaton () for the case f c/ s deterned by the functon values Ф(ξ ) at the pont ξ/ n the ddle of the nterval [ h] Slar to ths ntegratng polynoals () at the segents [ ] and [ /] the followng plct syste of n algebrac equatons can be obtaned to deterne the solutons at the ponts ξ / and ξ : ( ) ( ) () y + P() Ф/+ γ() Ф + ( ) η() Ф ( ) ( ) (/) y + P(/) Ф/ + γ(/) Ф + ( ) η(/) Ф where γ ξ ) I ( ξ ) P( ξ ) ψ (/ ) η ( ξ ) J ( ξ ) P( ξ ) χ (/ ) ( (5) 56
P( ξ ) ξ + + + + + + ξ ( ξ ) dξ ξ P() + ( ) ( + + ) ( + ) Herenafter the subscrpt of the coeffcent and rght-part functons ndcates an arguent value The soluton ethods of the auchy-proble for ODE-systes based on a step-to-step soluton of systes of algebrac equatons (5) are called by the M-ethods The estaton of the local error of the soluton at the step h: s + y( y( h The values generate an A-stable M-ethod of the 6th order [6] whose realzaton needs the calculatons of only frst dervatves of the syste of the rght-part functons () The calculaton schee of ths ethod conssts n solvng of the syste of n the algebrac equatons 3 3 () 4 9 7 () / y + Ф + Ф + Ф/ Ф + Ф 48 96 5 48 96 7 () 8 7 () y + Ф Ф + Ф/ + Ф Ф 3 6 5 3 6 Usng the equaton n varatons allows one n the gven case to obtan an estaton of the local error δ y y y at the step h and to accuulate the global error of the soluton For lnear and quas-lnear ODE-systes (wth thecoeffcent dependng on the arguent) the dervatves of the rght-part functons are lnear functons of the phase varables and therefore applcaton of schees (3) or (5) to such systes s splfed because n ths case systes (3) and (5) are systes of lnear algebrac equatons Ths crcustance s used to solve systes (3) and (5) by the ethod of sple teratons or by the Newton ethod for obtanng the frst approxaton of the soluton for whch the lnearzed ntal syste of ODE s used MD-ethod As t s shown for the - ethods - stablty s reached f one uses ore nforaton at the rght part of the ntegraton nterval than at a left one However to calculate the Hessan rght-part functons t s necessary to generate rather coplex algorths Below a varant of the M-ethod wth s proposed usng the values of the rght-part functons near the rght end of the ntegraton nterval at the pont ξ << Thus we obtan the ethod of 7 th order of accuracy n ters of the step h whch has both А- and -stablty; t does not requre calculaton of Hessans of the rght-part functons of ODE The rght-part functons of syste () has the for Ф ( ξ) ( ξ) ( a + aξ) + ξ [ b + b( ξ)] + + ( ξ) ξ [ c/ ( ζ) + c ζ] ζ (/ ξ) /(/ ξ ) ξ To deterne the coeffcents the followng relatonshps are obtaned: a Ф a Ф + Ф b Ф b Ф Ф c 6Ф 8( Ф + Ф ) + ( Ф Ф ) c / / ' ' (3 )( Ф Ф) ( ) ( ) Ф Ф Ф Ф + + Φ + ( ) ( ) (6) (7) 57
Snce we have () Ф Ф Ф + Ф + o( ) l c 3( Ф Ф ) Ф Ф + Ф Integratng syste () wth functons of the rght parts (6) we obtan the followng calculaton schee: / y + (7 / 4) a + (/9) a + (/ 4) b + (5/9) b + 5 6 7 + c/ + c 9( ) 96( ) y + α ( ξ ) a + α ( ξ ) a + α ( ξ ) b + α ( ξ ) b+ (8) + α ( ξ ) c/+ ( α3 ( ξ ) α ( ξ ) / )( c c/ ) /( ξ / ) y y + (/ 3) ( a + b ) + (/)( a + b ) + (/ 3) c () / / The one-step ethod usng the soluton at the step of the algebrac syste (8) was called the MD-ethod It s shown that the MD-ethod gves a soluton of the 7 th order of accuracy has A-stablty at and-stablty To solve the syste of algebrac equatons (8) a ethod of sple teratons s used whose convergence s provded by an approprate choce of the ntegraton step h [7] To deterne the frst approxaton of the soluton the M-ethod s used wth the paraeters The correspondng syste of equatons obtaned fro (5) has the for: / y + (5/ 4) Ф + (/ 3) Ф/ + ( / 4) Ф (9) y + (/ 6) Ф + ( / 3) Ф + (/ 6) Ф / The syste of equatons (8) s solved by the Newtonethod; the soluton obtaned has an errorofthe4 th order fro the value of the ntegraton step whch s rather good approxaton of the soluton y ( ξ ) EFEENES Berezn IS Zhdov NP alculaton Methods Мoscow: Naua 97 Nulchev YuV Nuercal ethod of ntegraton of systes of ordnary dfferental equatons based on analytcal dfferentaton of functons usng a coputer // Nuercal Methods of ontnuu Mechancs Vol No Novosbrs 98 P 33-4 (n ussan) 3 Dahlqust G A specal stablty proble for lnear ultstep ethods // BIT 963 Vol 3 P 7-43 4 Baer and GA Graves-Morrs P Pade Approxatons ondon et al: Addson-Wesley Publ o 98 [Translated fro Englsh Moscow: Mr 985] 5 Wanner G Harer E Norsett SP Order stars and stablty theores // BIT 978 Vol 8 P 475-489 6 Aulcheno SM atypov AF Nulchev YuV The ethod of nuercal ntegraton of the syste of ordnary dfferental equatons usng Hert nterpolaton polynoals // J op Math Math Physcs 998 Vol 38 No P 665-67 7 IG Petrovsy ectures on theory of ordnary dfferental equatons Moscow Naua 97 58