Applications of Mathematics

Transcription

1 Applications of Mathematics Zdzisław Jackiewicz; Rossana Vermiglio Order conditions for partitioned Runge-Kutta methods Applications of Mathematics Vol ) No Persistent URL: Terms of use: Institute of Mathematics AS CR 2000 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use Each copy of any part of this document must contain these Terms of use This paper has been digitized optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 ) APPLICATIONS OF MATHEMATICS No ORDER CONDITIONS FOR PARTITIONED RUNGE-KUTTA METHODS Zdzis law Jackiewicz TempeRossana Vermiglio Udine Received October 2 998) Abstract We illustrate the use of the recent approach by P Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations Keywords: Partitioned Runge-Kutta method ordinary differential equation order conditions MSC 2000 : 65L05 65L06 Introduction Consider a partitioned system of ordinary differential equations ODEs) ) { p = f p t p q) q = f q t p q) t [t 0 T] p R d q R d2 where the functions f p and f q are sufficiently smooth Such systems are encountered in many practical applications Typical examples are second order differential equations y = ft y y ) which by putting p = y and q = y canbewrittenintheform { p = q q = ft p q); Hamiltonian systems p = H q q =+ H p 30

3 where d = d 2 = d is the number of the degrees of freedom p are generalized coordinates q are generalized momenta and H is the Hamiltonian total energy of the system); and singular perturbation problems { y = fy z) εz = gy z) where ε is a small parameter The Volterra integral equation of the second kind t yt) =gt)+ k t s ys) ) ds t 0 t [t 0 T] can also be written as an infinite system of partitioned ODEs as demonstrated by Brunner Hairer and Nørsett [4] The system ) will be solved by the partitioned Runge-Kutta PRK) method 2) P i = p n + h Q i = q n + h q n+ = q n + h s a ij f p t n+cj P j Q j ) s ã ij f q t n+ cj P j Q j ) p n+ = p n + h s b i f p t n+ci P i Q i ) i= s bi f q t n+ ci P i Q i ) i= i = 2s n =0 N Nh = T t 0 wherethecoefficientsc i a ij b i and c i ã ij b i c i = s a ij c i = s ã ij represent two different RK schemes They will be represented by two tables c A b T = c c s a a s a s a ss c à bt c = c s ã ã s ã s ã ss b b s b b s Hairer [6] see also [7]) developed a theory of P -series to study order conditions for 2) As observed in [7] the derivation of order conditions simplifies considerably if we assume that c = c The resulting order conditions in this case up to the order r = 4 have been listed in Griepentrog [5] However in many practical applications 302

4 it is important to consider methods with c c This is the case for example when integrating separable Hamiltonian systems ie systems for which the Hamiltonian H has the special structure Ht p q) =T p)+v t q) In applications to mechanics T and V would represent the kinetic and the potential energy of the system respectively It is known see for example []) that such systems can be efficiently integrated by symplectic PRK methods which are also effectively explicit These methods have the form c 2 b c 3 b b c s b b 2 b 3 0 b b 2 b 3 b s c b c 2 b b2 0 0 c 3 b b2 b3 0 c s b b2 b3 b s b b2 b3 b s c i = i b j c i = i b j i = 2s hence in general c c It is the purpose of this paper to derive order conditions for PRK methods with c c using the alternative approach to the order theory developed by Albrecht [2] [3] This approach was also used to derive order conditions for Rosenbrock methods by Albrecht [2] for constant and variable two-step RK methods by Jackiewicz and Tracogna [9] [0] and for a certain class of general linear methods by Jackiewicz and Vermiglio [8] As in the case of the approach based on the P -series [6] [7] Albrecht s approach also simplifies considerably for PRK methods with c = c Introducing the notation 2 General order condition P =[P P s ] T t n+c =[t n+c t n+cs ] T t n+ c =[t n+ c t n+ cs ] T Q =[Q Q s ] T t n+ci = t n + c i h t n+ ci = t n + c i h f p t n+c PQ)=[f p t n+c P Q )f p t n+cs P s Q s )] T f q t n+ c PQ)=[f q t n+ c P Q )f q t n+ cs P s Q s )] T e =[] T R s 303

5 the method 2) can be written in the matrix form P = p n e + haf p t n+c PQ) Q = q n e + hãf q t n+ c PQ) p n+ = p n + hb T f p t n+c PQ) q n+ = q n + h b T f q t n+ c PQ) n =0 N We define the local discretization errors hd p n+c hd q n+ c h ˆd p n+ h ˆd q n+ of stage values P Q and the last stages p n+ q n+ as residua obtained by replacing in 2) the values P Q by pt n+c ) qt n+ c )andp n+ q n+ by pt n+ ) qt n+ ) where { ptn+c )=[pt n+c)pt n+cs)] T qt n+ c )=[qt n+ c )qt n+ cs )] T This leads to pt n+c )=pt n )e + haf p t n+c pt n+c )qt n+ c ) ) + hd p n+c qt n+ c )=qt n )e + 22) hãf q t n+ c pt n+c )qt n+ c ) ) + hd q n+ c pt n+ )=pt n )+hb T f p t n+c pt n+c )qt n+ c ) ) + h ˆd p n+ qt n+ )=qt n )+h b T f q t n+ c p t n+c )qt n+ c ) ) + h ˆd q n+ n =0 N Putting ξ p n+c = pt n+c ) P ˆξ p n+ = pt n+) p n+ ξ q n+ c = qt n+ c) Q ˆξ q n+ = qt n+) q n+ η p n+c = f p t n+c pt n+c )qt n+ c ) ) f p t n+c PQ) η q n+ c = f q t n+ c pt n+c )qt n+ c ) ) f q t n+ c PQ) and subtracting 2) from 22) we obtain 304 ξ p n+c = ˆξ n p e + haηp n+c + hdp n+c ξ q n+ c = ˆξ ne q + hãηq n+ c + hdq n+ c ˆξ p n+ = ˆξ n p + hb T η p n+c + h ˆd p n+ = ˆξ p n + hbt η p n +c + h ˆd p n + hbt η p n+c + h ˆd p n+ n = = h ˆd p l+ + n hbt η p l+c l=0 ˆξ q n+ = ˆξ q n + h b T η q n+ c + h ˆd q n+ l=0 = ˆξ q n + h b T η q n + c + h ˆd q n + h b T η q n+ c + h ˆd q n+ n = = h ˆd q l+ + h b n T l=0 l=0 η q l+ c

6 since ˆξ p 0 = pt 0) p 0 =0andˆξ q 0 = qt 0) q 0 = 0 The above relations lead to the following general order condition for PRK method 2) Theorem 2 Assume that the external stages p n+ and q n+ of the PRK method 2) have order of consistency r 3 ie ˆdp n = Oh r ) and ˆd q n = Ohr ) n =0 N Then the PRK method 2) converges with order r ie ˆξ p n = Oh r ) and ˆξ q n = Ohr ) n =0 N if and only if 23) { b T η p n+c = Oh r ) bt η q n+ c = Oh r ) n =0 N Moreover the global errors ξ p n+c and ξ q n+ c of the internal stages P and Q are given by 24) { ξ p n+c = hdp n+c + haηp n+c + Ohr ) ξ q n+ c = hdq n+ c + hãηq n+ c + Ohr ) n =0 N wherehd p n+c and hd q n+ c are defined in 22) Proof It follows from the assumptions of the theorem and from 23) that ˆξ p n+ = Ohr )andˆξ q n+ = Ohr ) which means convergence of order r Substituting the above relations into the formulas for ξ p n+c and ξ q n+ c we obtain 24) The order conditions 23) are not convenient to use since they depend on the functions f p and f q appearing in ) through η p n+c and η q n+ c ) In the rest of the paper we will reformulate 23) in terms of the coefficients c A b and c à b of PRK method 2) 3 Reformulation of order conditions for PRK methods The following technical lemma expresses η p n+c and ηq n+ c in terms of ξp n+c and ξ q n+ c This lemma will be instrumental in expressing order conditions 23) in a more convenient form 305

7 Lemma 3 Assume that the functions p and q are sufficiently smooth Then 3) η p n+c = η q n+ c = l= ν= l ) l+ l= ν= l! ) l l ν µ= j! qj) t n ) D D) j h j l ) l+ l! ) l l ν µ= j! pj) t n )D D) j h j µ! Gp lνµ t n) ) µ ξn+c) p l ν ξ q n+ c )ν µ! Gq lνµ t n) ) µ ξn+c) p l ν ξ q n+ c )ν with 32) where 33) and G p lνµ t) :=diag g p lνµ t + c h)g p lνµ t + c sh) ) =g p lνµ t)i + hdgp lνµ ) t)+ 2 h2 D 2 g p lνµ ) t)+ G q lνµ t) :=diag g q lνµ t + c h)g q lνµ t + c sh) ) =g q lνµ t)i + h Dg q lνµ ) t)+ 2 h2 D2 g q lνµ ) t)+ g p lνµ t) := l+µ f p p l ν q ν+µ t pt)qt) ) g q lνµ t) := l+µ f q ) t pt)qt) p l ν+µ q ν D = diagc c s ) D = diag c c s ) Proof Expanding f p t n+ci P i Q i )andf q t n+ ci P i Q i ) into Taylor series around t n+ci pt n+ci )qt n+ ci ) ) and t n+ ci pt n+ci )qt n+ ci ) ) respectively we obtain η p n+c i = η q n+ c i = l ) l+ ) l l! l ν µ! gp lνµ t n+c i ) l= ν=0 µ=0 qt n+ ci ) qt n+ci ) ) µ ξ p n+c i ) l ν ξ q n+ c i ) ν l ) l+ ) l l! l ν µ! gq lνµ t n+ c i ) l= ν=0 µ=0 pt n+ci ) pt n+ ci ) ) µ ξ p n+c i ) l ν ξ q n+ c i ) ν 306

8 where g p lνµ and gq lνµ are defined by 33) Substituting and qt n+ ci ) qt n+ci )= pt n+ci ) pt n+ ci )= j! qj) t n ) c i c i ) j h j j! pj) t n )c i c i ) j h j into η p n+c i and η q n+ c i we obtain formulas which are equivalent to 3) with G p lνµ and defined by 32) G q lνµ Using the notation employed in the above lemma the local discretization errors hd p n+c hd q n+ c h ˆd p n+ and h ˆd q n+ appearing in 22) of the PRK method 2) can be writtenintheform r+ hd p n+c = γ p l hl p l) t n ) 34) l= ha µ! Gp 00µ t ) µ n) j! qj) t n ) c c) j h j µ= +Oh r+2 ) r+ hd q n+ c = γ q l hl q l) t n ) l= hã µ= +Oh r+2 ) µ! Gq 00µ t ) µ n) j! pj) t n )c c) j h j 35) h ˆd r+ p n+ = ˆγ p l hl p l) t n ) l= hb T µ! Gp 00µ t ) µ n) j! qj) t n ) c c) j h j µ= +Oh r+2 ) h ˆd r+ q n+ = ˆγ q l hl q l) t n ) l= h b T µ= +Oh r+2 ) µ! Gq 00µ t ) µ n) j! pj) t n )c c) j h j 307

9 where γ p l := cl l! A cl l )! ˆγ p l := l! c l bt l )! l = 2 Define functions u p u q v p and v q by γ q l := cl l! Ã cl l )! ˆγ q l := l! b T c l l )! u p := ξ p n+c hd p n+c haη p n+c + Oh r ) u q := ξ q n+ c hdq n+ c hãηq n+ c + Ohr ) l v p := η p ) l+ ) l n+c l! l ν l= ν= µ= j! qj) t n ) D D) j h j v q := η q n+ c l= ν= l ) l+ ) l l! l ν j! pj) t n )D D) j h j µ! Gp lνµ t n) ) µ ξ p n+c) l ν ξ q n+ c )ν µ= µ! Gq lνµ t n) ) µ ξ p n+c) l ν ξ q n+ c )ν These functions are analytical at ξ p n+c =0ξ q n+ c =0ηp n+c =0η q n+ c =0andh =0 moreover det u p ξ p n+c u q ξ p n+c v p ξ p n+c v q ξ p n+c u p ξ q n+ c u q ξ q n+ c v p ξ q n+ c v q ξ q n+ c u p η p n+c u q η p n+c v p η p n+c v q η p n+c u p η q n+ c u q η q n+ c v p η q n+ c v q η q n+ c I =det 0 I 0 0 G p 00 G q 0 I 0 = G q 00 G q 0 0 I ξ p n+c ξq n+ c ηp n+c ηq n+ c h)=00000) Therefore it follows from the implicit function theorem that the system of equations 308 u p =0 u q =0 v p =0 v q =0

10 has a unique solution ξn+cξ p q n+ c ηp n+cη q n+ c ) around h = 0 This solution can be expanded into the Taylor series of the form 36) { ξ p n+c = sp 2 t n)h 2 + s p 3 t n)h s p r t n)h r + Oh r ) ξ q n+ c = sq 2 t n)h 2 + s q 3 t n)h s q r t n)h r + Oh r ) and 37) { η p n+c = w p 2 t n)h 2 + w p 3 t n)h w p r t n)h r + Oh r ) η q n+ c = wq 2 t n)h 2 + w q 3 t n)h w q r t n)h r + Oh r ) The fact that the above expressions start with terms of order Oh 2 ) follows from the definitions of ξ p n+c ξq n+ c ηp n+c and ηq n+ c Using the formulas for h ˆd p n+ h ˆd q n+ compare 35)) and the expressions 37) we can reformulate Theorem 2 in a more convenient form as follows Theorem 32 The PRK method 2) has order r if and only if ˆd p n = Oh r ) ˆd q n = Ohr ) n =0 Nand 38) { b T w p i = Ohr ) bt w q i = Ohr ) i =2 3r 4 Recursive generation of order conditions We will need the multinomial formula 4) where k= ) m x k k! tk = m! n=m t n n! n; a a 2 a n ) := a +2a 2++na n=n a +a 2++a n=m n; a a 2 a n ) x a n!!) a a!2!) a2 a 2! n!) an a n! xa2 n 2 xan compare [] page 823 We have the following theorem 309

11 Theorem 4 The vectors r p i rq i wp i andwq i appearing in 36) and 37) satisfy the recursions 42) 43) 44) r p i = γp i pi) + Aw p i a +2a 2++γa γ=γ a +a 2++a γ=µ r q i = γq i qi) + Aw q i w p i = a +2a 2++γa γ=γ a +a 2++a γ=µ l= ν= l ) l+ l! a +2a 2++γa γ=γ a +a 2++a γ=µ µ= δ+γ+=i δ 0 γ µ δ! Dδ c c) γ g p 00µ )δ) q ) a q ) a2 q γ) ) aγ!) a a!2!) a2 a 2! γ!) aγ a γ! µ= δ+γ+=i δ 0 γ µ δ! D δ c c) γ g q 00µ )δ) p ) a p ) a2 p γ) ) aγ!) a a!2!) a2 a 2! γ!) aγ a γ! l l ν ) µ=0 α ++α l ν +β ++β ν+δ+γ=i α α l ν β β ν 2 0 δ+γ i 2l δ 0 γ µ!) a a!2!) a2 a 2! γ!) aγ a γ! δ! Dδ D D) γ g p lνµ )δ) q ) a q γ) ) aγ r p α r p α l ν r q β r q β ν and 45) w q i = l= ν= l ) l+ l! a +2a 2++γa γ=γ a +a 2++a γ=µ ) l l ν µ=0 α ++α l ν +β ++β ν+δ+γ=i α α l ν β β ν 2 0 δ+γ i 2l δ 0 γ µ!) a a!2!) a2 a 2! γ!) aγ a γ! δ! D δ D D) γ g q lνµ )δ) p ) a p γ) ) aγ r p α r p α l ν r q β r q β ν w p := 0 wq := 0 i =2 3r 30

12 Proof Substituting 34) 36) and 37) into 24) and using 4) we obtain and s p 2 h2 + s p 3 h3 + + s p i hi + = γ p 2 h2 + γ p 3 h3 + + γ p i hi + + haw p 2 h2 + w p 3 h3 + + w p i hi + ) D δ c c) γ ha δ!γ! µ= i=µ+ δ+γ+=i δ 0 γ µ a +2a 2++γa γ=γ a +a 2++a γ=µ γ; a a 2 a γ ) g p 00µ )δ) q ) a q ) a2 q γ) ) aγ h i s q 2 h2 + s q 3 h3 + + s q i hi + = γ q 2 h2 + γ q 3 h3 + + γ q i hi + + hãwq 2 h2 + w q 3 h3 + + w q i hi + ) D δ c c) γ hã δ!γ! µ= i=µ+ δ+γ+=i δ 0 γ µ a +2a 2++γa γ=γ a +a 2++a γ=µ γ; a a 2 a γ ) g q 00µ )δ) p ) a p ) a2 p γ) ) aγ h i Comparing the terms corresponding to h i we obtain 42) and 43) Substituting 36) and 37) into 3) we have also w p 2 h2 + w p 3 h3 + + w p i hi + l ) l+ ) l = g p lνµ l! l ν µ! I + hdgp lνµ ) + 2 h2 D 2 g p lνµ ) + ) l= ν=0 µ! D D) γ h γ γ=µ γ! µ=0 a +2a 2++γa γ=γ a +a 2++a γ=µ γ; a a 2 a γ ) q ) a q γ) ) aγ r p 2 h2 + r p 3 h3 + ) l ν r q 2 h2 + r q 3 h3 + ) ν l ) l+ ) l = g p lνµ l! l ν I + hdgp lνµ ) + 2 h2 D 2 g p lνµ ) + ) l= ν=0 D D) γ h γ γ=µ γ! α α l ν 2 µ=0 a +2a 2++γa γ=γ a +a 2++a γ=µ r p α r p α l ν h α++α l ν γ; a a 2 a γ ) q ) a q γ) ) aγ β β ν 2 r p β r p β ν h β++βν 3

13 and w q 2 h2 + w q 3 h3 + + w q i hi + l ) l+ ) l = g q lνµ l! l ν µ! I + h Dg q lνµ ) + 2 h2 D2 g q lνµ ) + ) l= ν=0 µ! D D) γ h γ γ=µ γ! µ=0 a +2a 2++γa γ=γ a +a 2++a γ=µ γ; a a 2 a γ ) p ) a p γ) ) aγ r p 2 h2 + r p 3 h3 + ) l ν r q 2 h2 + r q 3 h3 + ) ν l ) l+ ) l = g q lνµ l! l ν I + h Dg q lνµ ) + 2 h2 D2 g q lνµ ) + ) l= ν=0 D D) γ h γ γ=µ γ! α α l ν 2 µ=0 a +2a 2++γa γ=γ a +a 2++a γ=µ r p α r p α l ν h α++α l ν γ; a a 2 a γ ) p ) a p γ) ) aγ β β ν 2 r p β r p β ν h β++βν Comparing the terms corresponding to h i we obtain formulas 44) and 45) The relations 42) 43) 44) and 45) simplify considerably if c = c compare also the comment in [7] about the approach based on P -series) In this case these relations assume the form r p i = γp i pi) + Aw p i r q i = γq i qi) + Aw q i w p i = i 2l 32 w q i = l= ν=0 δ=0 α ++α l ν +β ++β ν+δ=i α α l ν β β ν 2 i 2l l= ν=0 δ=0 α ++α l ν +β ++β ν+δ=i α α l ν β β ν 2 ) l+ l! ) l+ l! ) l D δ l ν δ! gp lν )δ) rα p rα p l ν r q β r q β ν ) l D δ l ν δ! gq lν )δ) rα p rα p l ν r q β r q β ν

14 w p := 0 wq := 0 i =2 3r where g p lν t) := g q lν t) := l f p p l ν q ν t pt)qt) ) l f q p l ν q ν t pt)qt) ) 5 Computation of order conditions for PRK method of order four In this section we will illustrate the use of Theorem 4 to generate the conditions up to the order four for PRK method 2) Routine calculations lead to r p 2 = γp 2 p A c c)g p 00 q r q 2 = γq 2 q Ãc c)gq 00 p w p 2 = γp 2 gp 00 p A c c)g p 00 gp 00 q + γ q 2 gp 0 q Ãc c)gp 0 gq 00 p w q 2 = γp 2 gq 00 p A c c)g q 00 gp 00 q + γ q 2 gq 0 q Ãc c)gq 0 gq 00 p r p 3 = γp 3 p3) 2 A c c)2 g p 00 q AD c c)g p 00 ) q 2 A c c)2 g p 002 q ) 2 + Aγ p 2 gp 00 p A 2 c c)g p 00 gp 00 q + Aγ q 2 gp 0 q AÃc c)gp 0 gq 00 p r q 3 = γq 3 q3) 2Ãc c)2 g q 00 p à Dc c)g q 00 ) p 2 Ãc c)2 g q 002 p ) 2 + Ãγp 2 gq 00 p ÃA c c)gq 00 gp 00 q + Ãγq 2 gq 0 q Ã2 c c)g q 0 gq 00 p w p 3 = γp 3 gp 00 p3) 2 A c c)2 g p 00 gp 00 q AD c c)g p 00 gp 00 ) q 2 A c c)2 g p 00 gp 002 q ) 2 + Aγ p 2 gp 00 )2 p A 2 c c)g p 00 )2 g p 00 q + Aγ q 2 gp 00 gp 0 q AÃc c)gp 00 gp 0 gq 00 p + Dγ p 2 gp 00 ) p DA c c)g p 00 ) g p 00 q + D D)γ p 2 gp 0 q p D D)A c c)g p 0 q g p 00 q + γ q 3 gp 0 q3) 2 Ãc c)2 g p 0 gq 00 p à Dc c)g p 0 gq 00 ) p 2Ãc c)2 g p 0 gq 002 p ) 2 + Ãγp 2 gp 0 gq 00 p ÃA c c)gp 0 gq 00 gp 00 q + Ãγq 2 gp 0 gq 0 q Ã2 c c)g p 0 gq 0 gq 00 p + Dγ q 2 gp 0 ) q DÃc c)gp 0 ) g q 00 p + D D)γ q 2 gp q q D D)Ãc c)gp q g q 00 p 33

15 w q 3 = γp 3 gq 00 p3) 2 A c c)2 g q 00 gp 00 q AD c c)g q 00 gp 00 ) q 2 A c c)2 g q 00 gp 002 q ) 2 + Aγ p 2 gq 00 gp 00 p A 2 c c)g q 00 gp 00 gp 00 q + Aγ q 2 gq 00 gp 0 q AÃc c)gq 00 gp 0 gq 00 p + Dγ p 2 gq 00 ) p DA c c)g q 00 ) g p 00 q +D D)γ p 2 gq 0 p p D D)A c c)g q 0 p g p 00 q + γ q 3 gq 0 q3) 2Ãc c)2 g q 0 gq 00 p à Dc c)g q 0 gq 00 ) p 2 Ãc c)2 g q 0 gq 002 p ) 2 + Ãγp 2 gq 0 gq 00 p ÃA c c)gq 0 gq 00 gp 00 q + Ãγq 2 gq 0 )2 q Ã2 c c)g q 0 )2 g q 00 p + Dγ q 2 gq 0 ) q DÃc c)gq 0 ) g q 00 p +D D)γ q 2 gq p q D D)Ãc c)gq p g q 00 p It follows from Theorem 32 that PRK method 2) has order four if and only if ˆd p n = Oh r ) ˆd q n = Oh r ) n =0 N b T w p i = Oh4 )and b T w q i = Oh4 ) i =2 3 This leads to the following equations expressed only in terms of the coefficients c A b and c à b of PRK method 2) Order : b T e = bt e =; Order 2: b T c = 2 bt c = 2 b T c c) =0 bt c c) =0; Order 3: b T c 2 = 3 bt c = 3 b T c c) 2 =0 b T D c c) =0 bt c c) 2 =0 bt Dc c) =0 b T γ p 2 =0 b T A c c) =0 bt γ p 2 =0 A c c) =0 b T γ q 2 =0 b T Ãc c) =0 bt γ q 2 =0 Ãc c) =0; Order 4: b T c 3 = 4 bt c 3 = 4 b T c c) 3 =0 bt c c) 3 =0 b T D c c) 2 =0 bt Dc c) 2 =0 b T D 2 c c) =0 bt D2 c c) =0 b T γ p 3 =0 bt γ p 3 =0 b T A c c) 2 =0 bt A c c) 2 =0 b T AD c c) =0 bt AD c c) =0 b T Aγ p 2 =0 bt Aγ p 2 =0 b T A 2 c c) =0 bt A 2 c c) =0 b T Aγ q 2 =0 bt Aγ q 2 =0 34

16 T b AÃc c) =0 bt AÃc c) =0 b T Dγ p 2 =0 p bt Dγ 2 =0 b T DA c c) =0 bt DA c c) =0 b T D D)γ p 2 =0 bt D D)γ p 2 =0 b T D D)A c c) =0 bt D D)A c c) =0 b T γ q 3 =0 bt γ q 3 =0 b T Ãc c) 2 =0 bt Ãc c) 2 =0 b T à Dc c) =0 bt à Dc c) =0 b T Ãγ p 2 =0 bt Ãγ p 2 =0 b T ÃA c c) =0 bt ÃA c c) =0 b T Ãγ q 2 =0 bt Ãγ q 2 =0 b T à 2 c c) =0 bt à 2 c c) =0 b T Dγ q 2 =0 q bt Dγ 2 =0 b T DÃc c) =0 bt DÃc c) =0 b T D D)γ q 2 =0 bt D D)γ q 2 =0 b T D D)Ãc c) =0 bt D D)Ãc c) =0 In the case c = c these conditions reduce to the following much smaller set of equations listed below Order : b T e = bt e =; Order 2: b T c = 2 bt c = 2 ; Order 3: b T c 2 = 3 bt c = 3 b T γ p 2 =0 bt γ p 2 =0 b T γ q 2 =0 bt γ q 2 =0; Order 4: b T c 3 = 4 bt c 3 = 4 b T γ p 3 =0 bt γ p 3 =0 b T Aγ p 2 =0 bt Aγ p 2 =0 b T Aγ q 2 =0 bt Aγ q 2 =0 b T Dγ p 2 =0 bt Dγ p 2 =0 b T γ q 3 =0 bt γ q 3 =0 b T Ãγ p 2 =0 bt Ãγ p 2 =0 b T Ãγ q 2 =0 bt Ãγ q 2 =0 b T Dγ q 2 =0 bt Dγ q 2 =0 These relations are equivalent to the order conditions listed in [5] Acknowledegement This work was initiated during the visit of the first author to the University of Udine in May 997 This author wishes to express his gratitude to Professor R Vermiglio for making this visit possible 35

17 References [] M Abramowitz I S Stegun eds): Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables National Bureau of Standards Applied Mathematics Series 55 Washington DC 972 [2] P Albrecht: A new theoretical approach to Runge-Kutta methods SIAM J Numer Anal ) [3] P Albrecht: The Runge-Kutta theory in a nutshell SIAM J Numer Anal ) [4] H Brunner E Hairer and S P Nørsett: Runge-Kutta theory for Volterra integral equations of the second kind Math Comp ) [5] E Griepentrog: Gemischte Runge-Kutta-Verfahren für steife Systeme In: Seminarbericht Nr Sekt Math Humboldt Univ Berlin 978 pp 9 29 [6] E Hairer: Order conditions for numerical methods for partitioned ordinary differential equations Numer Math 36 98) [7] E Hairer S P Nørsett and G Wanner: Solving Ordinary Differential Equations I Nonstiff Problems Springer-Verlag Berlin-Heidelberg-New York 99 [8] Z Jackiewicz R Vermiglio: General linear methods with external stages of different orders BIT ) [9] Z Jackiewicz S Tracogna: A general class of two-step Runge-Kutta methods for ordinary differential equations SIAM J Numer Anal ) [0] Z Jackiewicz S Tracogna: Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations Numer Algorithms 2 996) [] J M Sanz-Serna M P Calvo: Numerical Hamiltonian Problems Chapman & Hall London-Glasgow-New York 994 Authors addresses: Zdzis law Jackiewicz Department of Mathematics Arizona State University Tempe Arizona USA jackiewi@mathlaasuedu; Rossana Vermiglio Department of Mathematics and Computer Science University of Udine I-3300 Udine Italy rossana@dimiuniudit 36