Order Results for Mono-Implicit Runge-Kutta Methods K urrage, F H hipman, and P H Muir bstract The mono-implicit Runge-Kutta methods are a subclass of the wellknown family of implicit Runge-Kutta methods and have application in the ecient numerical solution of systems of initial and boundary value ordinary dierential equations lthough the eciency and stability properties of this class of methods have been studied in a number of papers, the specic question of determining the maximum order of an s-stage mono-implicit Runge-Kutta method has not been dealt with In addition to the complete characterization of some subclasses of these methods having a small number of stages, and the presentation of some new methods from this class, a main result of this paper is a proof that the order of an s-stage mono-implicit Runge-Kutta method is at most s + Keywords: Runge-Kutta methods, ordinary dierential equations MS subject classication: 65L5 This work was supported by the Natural Sciences and Engineering Research ouncil of anada uthor addresses: Kurrage, entre for Industrial and pplied Mathematics and Parallel omputing, Department of Mathematics, The University of Queensland, risbane 472, ustralia E-mail: kbmathsuqozau; FHhipman, cadia University, Wolfville, Nova Scotia, anada, P X E-mail: chipmanacadiauca; PHMuir, Saint Mary's University, Halifax, Nova Scotia, anada, 3H 33 E-mail: muirhuskystmarysca)
Introduction Implicit Runge-Kutta (IRK) methods were rst presented in [4] for use in the numerical solution of initial value ordinary dierential equations Since that time there has been considerable attention devoted to these methods The reader is referred to the book [5] for extensive review of these methods n IRK method can be used to compute an approximation to the solution of the initial value problem y (t) = f(y(t)); y(t i ) = y i ; where y(t) 2 R n and f : R n! R n (For convenience we will consider only autonomous systems) Using an IRK method, we obtain an approximation, y i+, to the true solution, y(t), evaluated at the point t i+ = t i + h i, of the form, where ^y r = y i + h i y i+ = y i + h i sx j= sx r= b r f(^y r ); (:) a r;j f(^y j ); r = ; : : : ; s: (:2) These methods are often given in the form of a tableau containing their coecients, which, for the above method, would have the following form c a ; : : : a ;s c s a s; : : : a s;s b : : : b s where c i = P s j= a i;j Let b = (b ; b 2 ; : : :; b s ) T, be the s s matrix whose (i; j)-th component is a i;j, and e is the vector of 's of length s Then c = (c ; c 2 ; : : :; c s ) T, can be expressed as c = e, and the stability function of the IRK method is given by R(h) = + hb T (I? h)? e: 2
Note that in (2) above, each stage, ^y r, is dened implicitly in terms of itself and the other stages Hence in order to obtain approximate values for the stages it is necessary to solve a system of, in general, n s non-linear equations This is often done using some form of modied Newton iteration, which makes the calculation of the stages a somewhat computationally expensive process number of interesting subclasses of the IRK methods have recently been identied and investigated in the literature These methods represent attempts to trade-o the higher accuracy of the IRK methods for methods which can be implemented more eciently Examples of such methods are singly-implicit Runge-Kutta methods [2], diagonally implicit Runge-Kutta methods [2], and mono-implicit Runge-Kutta methods (see [,] for initial value problems and [,3,5] for boundary value problems) See [8] for a survey of these methods In [8], an alternative representation of the IRK methods known parameterized implicit Runge-Kutta (PIRK) methods was suggested These have the form, y i+ = y i + h i sx r= b r f(^y r ); (:3) where ^y r = (? v r )y i + v r y i+ + h i sx j= x r;j f(^y j ); r = ; : : : ; s: (:4) They are usually represented by a modied tableau; the above method would have the tableau, c v x ; : : : x ;s c s v s x s; : : : x s;s b : : : b s ; where c i = v i + P s j= x i;j Substituting for y i+ from (3) in (4), it is easily seen that the PIRK method given above is equivalent to the IRK method with coecient tableau 3
c x ; + v b x ;2 + v b 2 : : : x ;s + v b s c s x s; + v s b x s;2 + v s b 2 : : : x s;s + v s b s b b 2 : : : b s : Let v = (v ; v 2 ; : : :; v s ) T and X be the s s matrix whose (i; j)-th component is x i;j Then c = Xe + v and, from [8], the stability function, R(h), of a PIRK satises R(h) = P (h; e? v) P (h;?v) ; where P (h; w) = + hb T (I? hx)? w; w 2 R s : If we impose on the PIRK methods the restriction that the matrix X be strictly lower triangular then the resultant methods are known as mono-implicit Runge-Kutta (MIRK) methods MIRK methods have been discussed in the literature by a number of dierent authors for more than a decade subclass of these methods was suggested for use in the solution of initial value ODE problems in [6] The full MIRK class was presented, again for initial value problems, in [] (where they were called implicit endpoint quadrature formulas) and [] MIRK methods were proposed for boundary value ODE's in [7,8,9,,3,5,6] For VODE's, these methods can be implemented at a cost comparable to that of explicit Runge-Kutta methods (See [Enright and Muir 986]) MIRK methods have been used in the implementation of a software package, for the numerical solution of VODE's, using deferred corrections, called HGRON [8], [2] More recently, the MIRK methods have been used in a software package which employs continuous extensions of these methods (see [9]) to provide defect control for the numerical solution of VODE's, [4] Much of the above work has been concerned with the identication of particular MIRK formulas and the investigation of their ecient implementation s well, there has been some investigation of the forms of the order conditions for these methods, as they relate to the well-known utcher conditions for the standard IRK methods, in [] and [6] However, 4
a complete characterization of low order MIRK methods has not been undertaken, as has been done for low order explicit Runge-Kutta methods, [5] This characterization is important because it allows an analysis of the resultant families to determine new optimal schemes representing potential improvements of those currently presented in the literature Furthermore, the question of the maximum order of an s-stage MIRK has not been answered In [7], it was conjectured that this maximum order is s + and in [7] the maximum order question was identied as a signicant open problem in the study of the MIRK methods In Section 2 we present characterizations for several subclasses of MIRK methods having a small number of stages In Section 3 we present some general results on the maximum order of an s-stage MIRK method, and show that an upper bound for the maximum order is s + In Section 4 we give some conclusions and suggestions for future work 2 Low Order Mono-Implicit Runge-Kutta Methods In this section we present characterizations for a number of families of low order MIRK methods, having, 2, 3, 4, or 5 stages For the standard IRK method to have stage order p, it must satisfy the conditions (p) and (p) (see eg [5, Pg 289]), where (p) : b T c k? = =k; k = ; : : : ; p; (p) : c k? = c k =k; k = ; : : : ; p; where c k = (c k ; c k 2; : : :; c k s) For a MIRK method to have stage order p, it must satisfy the conditions, (p), as given above, and the following form of the (p) conditions, (p) : v + kxc k? = c k ; k = ; : : : ; p; since for a MIRK scheme expressed in standard IRK form, we have = X + vb T lso, for an IRK or MIRK method to be of classical order p +, it is sucient for the method to satisfy (p + ) and (p) See [5, Pg 24-29] It has recently become clear (see for example [5]) that in addition to the usual Runge- Kutta order concept, it is important to take into consideration the stage order of the Runge- Kutta method, at least in the context of solving sti systems of dierential equations The 5
basic idea is that a p-th order Runge-Kutta method with stage order q, (q < p) when applied to a sti ode system, can yield errors of order q or q +, rather than the expected order p s mentioned above, in searching for new methods it is important to require relatively high stage order Ideally for a p-th order method, we would like to require stage order p? In the discussion to follow, for families of MIRK methods having s stages, where s = ; 2; 3, we will present methods of order s + having stage order s, and methods of order s, having stage order s? For families of MIRK methods having 4-stages we will present 4-th order methods with stage order 3, but the following theorem shows that there is no point in looking for 5th order methods with stage order 4 Theorem 2: (i) a MIRK method having stage order 2 must have c = or, (ii) a MIRK method, with at least 2 stages and having stage order 3, must have x 2; = and either c = ; c 2 = or (equivalently) c = ; c 2 =, and (iii) the maximum stage-order of an s stage MIRK method is min(s; 3) Proof: (i) The stage order 2 conditions for a MIRK method are v+xe = c, and v+2xc = c 2 The rst equation from each of these gives v = c and v = c 2 from which it follows that c = or (ii) The stage order 3 conditions are v + Xe = c, v + 2Xc = c 2, and v + 3Xc 2 = c 3 From (i) we have either c = or c = If c = then the second equation from each of these is v 2 + x 2; = c 2, v 2 = c 2 2, and v 2 = c 3 2, from which it follows that x 2; =, and c 2 = so that the rst and second stages are not the same similar argument holds for the case where c = (iii) For s = or 2, an s-stage MIRK method can only satisfy, simultaneously, (a) (s+) and (s), or (b) (s) and (s+) In either case this implies that the stage order is s For s 3, suppose the MIRK method has stage order 4 Then it must satisfy the (4) conditions which are v + Xe = c, v + 2Xc = c 2, v + 3Xc 2 = c 3, and v + 4Xc 3 = c 4 gain from (i) we have either c = or c = For each value of c, it follows from (ii) that x 2; = and c 2 = if c =, or c 2 = if c =, so that the second stage is dierent from the rst Repeating this process for the third equation from each set of (4) conditions we come to 6
the conclusion that the third stage must be the same as either the rst or the second stage, thus implying that the MIRK has only distinct s? stages, a contradiction Hence the stage order of an s-stage MIRK method, with s 3, is at most 3 2 MIRK methods with stage For the class of stage MIRK methods a family of methods of order, including the explicit and implicit Euler methods, is obtained when b = ; v = c with c left as a free parameter If we choose c =, then we obtain the unique -stage MIRK of maximum order 2 2, namely the mid-point rule 22 MIRK methods with 2 stages Specic examples of order 2 MIRK methods are the trapezoidal rule, (x 2; =, c = v =, c 2 = v 2 =, b = b 2 = ), and a one parameter family of 2nd order 2-stage MIRK 2 methods given in [] (Method II) more interesting question involves looking for 3rd order MIRK methods with 2 stages n example of such a method is the 2-point Gauss-Radau IRK method (Radau II), given in MIRK form in [], (Method III) In order to look for 3rd order methods we must examine the corresponding order conditions for a 2-stage MIRK method It is a straightforward process to convert the standard utcher conditions for IRK methods (see [5, pg 234]) to modied utcher conditions that can be applied directly to methods expressed in terms of the MIRK tableau, ie with the inclusion of the extra column of parameters v ; v 2 ; : : :; v s For example, for 3rd order 2-stage MIRK methods we have the following conditions: the quadrature conditions, denoted as (3), b + b 2 = ; b c + b 2 c 2 = 2 ; b c 2 + b 2 c 22 = 3 ; and the condition, b 2 x 2; c + 2 (b v + b 2 v 2 ) = 6 : Solving the above system, we get the following parameter family of 2-stage 3rd order MIRK methods (letting c be the parameter), 7
c c 3c? 2 3(2c? ) 36c 3? 54c 2 + 27c? 4 9(2c? ) 3? 2(3c2? 3c + ) 9(2c? ) 3 4(3c 2? 3c + ) 3(4c 2? 4c + ) 4(3c 2? 3c + ) : >From an examination of the order conditions for order 4 MIRK schemes, it can easily be seen that it is impossible to choose the one free parameter, c, so that the above family can be of 4th order Thus, for 2-stage MIRK methods the maximum order is 3 The linear stability function associated with this family of methods is R(z) = 6(? 2c) + 2(2? 3c)z + (? c)z2 6(? 2c)? 2(? 3c)z? cz 2 ; and the method is -stable if and only if c > 2 lso, note that if c2? c + 6 holds This is the MIRK analogue of the 2-stage Gauss method of order 4 = then (4) The above family of methods, in general, has only stage order To nd third order methods having stage order 2, we must impose the (2) conditions on the family of third order methods above We nd that the only members that have stage order 2 are obtained by choosing c = or in the () family which gives, respectively, Method III (from []) mentioned above (which is -stable) and its reection [2] (which is not -stable) 23 MIRK methods with 3 stages We will rst look for 3rd order, 3-stage methods having stage order 2 To establish order 3, stage order 2, it is necessary and sucient to impose the (3) and (2) conditions Stage order 2 implies either c = or c = We consider the c = case, noting that its reection [2] will yield the c = case We get the following 3-parameter family of methods where c 2, c 3, and v 3 are the parameters, with the restriction that, c 2, and c 3 are all dierent c 2 c 2 (2? c 2 ) c 2 (c 2? ) c 3 v 3 2(? c 2 ) 2(c 2? ) c 3 (c 3? 2c 2 ) + v 3 (2c 2? ) v 3 + c 3 (c 3? 2) 6c 2 c 3? 3c 3? 3c 2 + 2 6(c 2? )(c 3? ) 3c 3? 6(c 2? )(c 2? c 3 ) 3c 2? 6(c 3? )(c 3? c 2 ) 8
The linear stability function is 6 + 2(? )z + (c R(z) = 2? )z 2 6? 2(2 + )z + ( + (c 2 + ))z 2? c 2 z 3 where = (c 3? )(c 3? c 2 ); and = 2 (v 3 + c 3 (c 3? 2))(3c 2? ): The methods are -stable if and only if > and (3c 2? ) < : If we were to require stageorder 3, thus making c 2 = x 2; =, v 3 = c 2 3(3? 2c 3 ), x 3; = c 2 3(c 3? ), and x 3;2 = c 3 (c 3? ) 2 2, the resulting method is -stable if and only if c 3 > The reection [2] of this method, with c =, can only be -stable if c 2 =, and hence x 2; = If we now look for 3-stage, stage-order 2 MIRK methods with order 4 we get a oneparameter family of methods with either c = or c = ; (a requirement for stage-order 2) onsidering the case c =, we have the following -parameter family (where the parameter is c is a real number, satisfying c 6= ;, or ): 4 3 c c(2? c) c(c? ) 2c? 6c? 2 (? 2c)(? 4c) 2(c? ) 6c 2? 6c + 6(4c? )(c? ) (4c? ) 2(c? )? 6(c? ) 2(3c? ) 3 3(4c? ) where = 8c 4? 24c 3 + 2c 2? 26c + 2, = 4(3c? ) 4, =? c + 24c 2? 8c 3, and = 6c 2? 4c + The linear stability function for this family of methods is given by: R(z) = P (z; c) Q(z; c) = 2(? 3c) + 6(? 2c)z + (? c)z 2 2(? 3c)? 6(? 4c)z + (? 7c)z 2 + cz 3 ; and the methods are -stable if and only if c > 3 stage-order 3 or c = The choice of c = gives The c = case gives the stability function, R(z) = Q(?z;? c) P (?z;? c) ; 9
and hence methods that are only -stable for the one choice c =, the stage-order 3 case It is easy to show that this stage order 3 method is the only such 4th order, 3-stage method It is given [] (Method IV) and is obtained by choosing c = v =, c 2 = v 2 =, x 2; =, c 3 = v 3 =, and x 2 3; =?x 3;2 = 8 24 MIRK methods with 4 stages s might be expected, is not dicult to obtain 4-stage MIRK methods of order 4 Two examples are given in [] (Methods V and VII), both of which have stage order 2 elow we give a 3-parameter family of 4-stage, 4-th order methods having stage order 3 These are obtained by imposing the (4) and (3) conditions The parameters are c 3, c 4, and v 4, with the restrictions that c 3 6= c 4, c 3 6= ; ;, and c 2 4 6= ; c 3 c 2 3 (3? 2c 3) c 3 (c 3? ) 2 c 2 3 (c 3? ) c 4 v 4 6c 3 6(c 3? ) 6c 3 (c 3? ) 2c 4?? 2c 3 2c 3 c 4 2(? c 3 )(? c 4 ) 2c 3 (c 4? c 3 )(? c 3 ) 2c 4 (c 4? c 3 )(? c 4 ) where = 6c 4 c 3 +v 4?3c 3 v 4?3c 3 c 2 4+2c 3 4?3c2 4, = 3c 3 (c 2 4?v 4)?2(c 3 4?v 4), =?2(c 3 +c 4 )+6c 3 c 4, = 3? 4(c 3 + c 4 ) + 6c 3 c 4, and = c 2 4(2c 4? 3) + v 4 The linear stability function is, R(z) = p + p z + p 2 z 2 + p 3 z 3 q + q z + q 2 z 2 + q 3 z 3 ; where, p = q = 2c 4 (c 4? )(c 4? c 3 ); p = 6c 4 (c 4? )(c 4? c 3 )? (c 2 4(2c 4? 3) + v 4 )(2c 3? ); q =?6c 4 (c 4? )(c 4? c 3 )? (c 2 4(2c 4? 3) + v 4 )(2c 3? ); p 2 = c 4 (c 4? )(c 4? c 3 ) + 3 (c2 4(2c 4? 3) + v 4 )(c 3? 2)(2c 3? ); q 2 = c 4 (c 4? )(c 4? c 3 ) + 3 (c2 4(2c 4? 3) + v 4 )(c 3 + )(2c 3? ); p 3 = 6 (c2 4(2c 4? 3) + v 4 )(2c 3? )(c 3? ); q 3 =? 6 (c2 4(2c 4? 3) + v 4 )c 3 (2c 3? ):
The family will be -stable if and only if c 3 > =2 and (2c 3? ) 6c 4 (c 4? )(c 4? c 3 ) > It is also possible to generate 4-stage, 5th order MIRK methods However from Theorem 2, it is only worthwhile to look for 4-stage, 5th order MIRK methods having at most stage order 3 elow we present a 4-stage, 5th order family of stage order 2, which includes a 4-stage, 5th order family of stage order 3 The stage order 2 family has 2 parameters, c 2 and c 3 c 2 c 2 (2? c 2 ) c 2 (c 2? ) c 3 c 3 (2? c 3 ) + 2x 3;2 (c 2? ) c 3 (c 3? )? x 3;2 (2c 2? ) x 3;2 5 5? x 4;? x 4;2? x 4;3 x 4; x 4;2 x 4;3? b 2? b 3? b 4 (c 3 ) (c 2 ) 25 4 (c 2 ; c 3 ) (c 3 ; c 2 ) 2 where = c 2 c 3? 5(c 2 + c 3 ) + 3, = 6c 2 c 3? 2(c 2 + c 3 ) +, = (5? )(5c 2? )(5c 3? ), (c) = c 2? 8c +, where c = c 2, or c 3, (c; d) = 2(? c)(d? c)(5c? ), where c; d = c 2 ; c 3, or c 3 ; c 2, and x 3;2 = (2c 2c 3 + c 2 + c 3? )(? c 3 )(c 2? c 3 ) ; (3c 2? )(c 2? )(c 2 ) x 4; = 5 ( 5? )? x (c 4;3(2c 3? )? x 4;2 (2c 2? ); x 4;3 = 2 ) 625 5 (c 3? )(c 2? c 3 ) ; x 4;2 = (5? )(5c 2? )(5c 3? 2)? 25 3 x 4;3 (3c 3? )(c 3? ) : 25 3 (3c 2? )(c 2? ) There are several restrictions on the values of the two parameters: c 2 6= c 3, c 2 6= 3 ;, c 3 6=, (c 2 ) 6=, 6=, 6=, (c 2 ; c 3 ) 6=, and (c 3 ; c 2 ) 6= The linear stability function for this family is given by R(z) = w + w z + w 2 z 2 + w 3 z 3 t? t z? t 2 z 2? t 3 z 3? t 4 z 4 ; where w = t = 6(3c 2? ); w = 2(3c 2? ); w 2 = 3(3c 2? ) + 6(? 2)(2c 2? ); w 3 = (? 2)(c 2? )
and t = 2(3c 2? )(5? ); t 2 = 6(? 2)(4c 2? )? 9(3c 2? ); t 3 = (3c 2? )? (7c 2? )(? 2); t 4 = c 2 (? 2): The methods are -stable if and only if < (2? )(2c 2 + ) (3c 2? ) < 2 one parameter 4-stage, 5th order, stage order 3 family, contained within the above family, is obtained simply by choosing c 2 = In this case, the methods are -stable if and only if c 3 > 25 MIRK methods with 5 stages We focus, in this section, on 5-stage methods of maximum order 6 n example of such a method is given in [] (Method VIII), where the stage order has the maximum value of 3 This method is generalized to a 5-stage, 6th order -parameter family of methods having stage order 3, in [6] (Table 2) In this section we complete the generalization of these methods by presenting a 2-parameter family of 5-stage, 6-th order methods having stage (? p) ( + p) order 3 The parameters are c 3 and c 4 ; by choosing c 3 = and c 2 4 = we get the 2 one parameter family (with parameter p) given in [6] The restrictions on the parameters c 3 and c 4 are c 3 6= ;, c 4 6= ;, 6= 6=, (c 3 ) 6=, (c 4 ) 6=, (c 3 ; c 4 ) 6=, (c 4 ; c 3 ) 6=, (c 3 ) 6=, and (c 4 ) 6= The 2-parameter family is given below c 3 c 2 3 (3? 2c 3) c 3 (c 3? ) 2 c 2 3 (c 3? ) c 4 c 2 4 (3? 2c 4) + 6x 4;3 c 3 (c 3? ) x 4; x 4;2 (c 3 + c 4? ) (c 4 ; c 3 ) 2(c 3 )(c 3 )? x 5;? x 5;2? x 5;3? x 5;4 x 5; x 5;2 x 5;3 x 5;4 b b 2 b 3 b 4 b 5 where x 4; = c 4 (c 4? ) 2? x 4;3 (3c 3? )(c 3? ); x 4;2 = c 2 4(c 4? )? x 4;3 (3c 3? 2)c 3 ; 2
b 3 = x 5; = c 5 (c 5? ) 2? x 5;3 (3c 3? )(c 3? )? x 5;4 (3c 4? )(c 4? ); x 5;2 = c 2 5(c 5? )? x 5;3 (3c 3? 2)c 3? x 5;4 (3c 4? 2)c 4 ; x 5;3 = 6b 5 (c 3 )? x b 4 (c 4 ) 4;3? x 5;4 b 5 (c 3 ) ; x 5;4 = (? )(c 3)(c 3 )(c 4 ) ; (c 4 ; c 3 ) 6 b = 5c2 3(c 2 4? 2c 4 + 3)? 5c 3 (2c 2 4? 5c 4 + 4) + (5c 2 4? 2c 4 + 6) ; 6(c 3? )(c 4? )(? ) b 2 = 5c2 3(c 2 4? 8c 4 + )? 5c 3 (8c 2 4? 7c 4 + ) + (c 4 ) ; 6c 3 c 4 (c 4 ) 6 (c 3 ; c 4 )(c 3 ) ; b 4 = (c 3 ) 6 (c 4 ; c 3 )(c 4 ) ; b 5 = 5 6(? )(c 3 )(c 4 ) ; and where = 5c 4 c 3? 3(c 3 + c 4 ) + 2, = c 3 c 4? 5(c 3 + c 4 ) + 3, (c) = 5c 2? 5c +, (c) = c(c? )(2c? ), (c) = c?, and (c; d) = c(c? )(c? d) The linear stability function is, R(z) = w + w z + w 2 z 2 + w 3 z 3 + w 4 z 4 t? t z? t 2 z 2? t 3 z 3? t 4 z 4 ; where w = t = 2(2c 3? ); w = 2(2c 3? ); t = 2(2c 3? )(5c 3 c 4? 2(c 3 + c 4 ) + ) w 2 = 2(2c 32 c 4? 25c 3 c 4 + 7c 4? 5c 2 3 + 8c 3? 5); t 2 =?2(2c 2 3c 4? 5c 3 c 4 + 2c 4? 5c 2 3 + 3c 3 ); w 3 = 2(c 2 3c 4? 6c 3 c 4 + 4c 4? c 2 3 + 3c 3? 3); t 3 = 2(c 32 c 4? 4c 3 c 4? 2c 4 + c 2 3? 5c 3 + 3); w 4 = c 3 (? c 3? c 4 ); t 4 = (c 3? )(? c 3? c 4 ): The methods are -stable if and only if c 3 < =2 and (? c 3? c 4 ) 3 The Maximum Order of n s-stage MIRK Method > In this section, we will present a proof giving an upper bound on the order of an s- stage MIRK scheme, as given in (3),(4), with coecients represented by the vectors, c = (c ; c 2 ; : : : ; c s ) T, v = (v ; v 2 ; : : : ; v s ) T, b = (b ; b 2 ; : : : ; b s ) T, and the strictly lower triangular s s matrix X, whose (i; j)-th component is x ij s before, e is the vector of 's of length s The proof is based on a subset of the Runge-Kutta order conditions, which we now present Lemma 3: For an s-stage, order s + 2 MIRK scheme b T X k?p (c p+? c) = b T X k?p+ ((p + )c p? e); p = ; ; : : :; s; k = p; : : :; s: (T ) 3
(We refer to these conditions as the "cabbage tree" conditions (T's), since they are based on the order conditions associated with a tree consisting of a trunk of i edges surmounted by a bunch of j edges) Proof: mong the order conditions that any Runge-Kutta scheme of order s+2 must satisfy are b T k?p+ c p = p!=(k + 2)! and b T k?p c p+ = (p + )!=(k + 2)!: Let t = c p+? (p + )c p : Then b T k?p t = ; p = ; ; : : :; s k = p; : : :; s; giving (for a MIRK scheme where = X + vb T ), b T k?p t = b T (X + vb T ) k?p? t = b T X k?p? t = b T X 2 k?p?2 t = : : : = b T X k?p t = : Since, for a MIRK scheme, v = c? Xe, we get, from which the result follows b T X k?p (c p+? (p + )X(c p? e )? c) = ; p + We note that since X is strictly lower triangular, the vector b T X i has at least i zero elements, in the last i positions We are interested in the (potentially) non-zero elements only, and so shall denote by y i = (y i; : : :; y i;i ) T, the vector of length i consisting of the rst i elements of b T X s?i With a k by k matrix, and V k? = V k = c 2? c c 2 2? c 2 : : : c 2 k? c k c 3? c c 3 2? c 2 : : : c 3? c k k c k+? c c k+ 2? c 2 : : : c k+ k? c k 2c? 2c 2? : : : 2c k?? 3c 2? 3c 2? : : : 3c 2? k? (k + )c k? (k + )c k 2? : : : (k + )c k k?? 4 ; ;
a k by k? matrix, the T's may be grouped and rewritten as, V y = ; () and nally, V 2 y 2 = V y ; (2) V 3 y 3 = V 2 y 2; (3) V s? y s? = V s?2 y s?2; (s? ) V s?y s? =? 2 =3 2=4 s=s + 2 ; (s) where we note that T(k) consists of k equations, k = : : : s In the remainder of this proof, the right hand sides of some of the T's will reduce to the zero vector When it arises in T(k), will denote the zero vector of length k In the proof of the main result that follows, we will need two lemmas Lemma 32: Suppose the matrix V k? is non-singular and c 2 k = c k Then the matrix, is nonsingular ^V k = c 2? c : : : c 2 k?? c k? 2c k? c k+? c : : : c k+ k?? c k? (k + )c k k? Proof: Let f(c k ) = det(v k ) and note that f has simple zeros at c k = and c k = Hence f (c k ) is non-zero at c k = ; ut f (c k ) is just det( ^Vk ), proving the result ; Lemma 33: Suppose that the abscissa, fc i g; i = ; : : :; l, are distinct, and that the matrix V k? is non-singular Further suppose that c 2 k = c k, c 2 6= c j j; j = k + ; : : : ; l?, and c 2 l = c l Then the matrix, ^V l, given by, 5
c 2? c : : : c 2 k?? c k? 2c k? c 2 k+? c k+ : : : c 2 l?? c l? 2c l? c l+? c : : : c l+ k?? c k? (k + )c l k? cl+ k+? c k+ : : : c l+ l?? c l? (l + )c l l? is nonsingular Proof: Let f(c k ; c l ) = det(v l ) and note that f has simple zeros at c k = or and c l = or Hence 2 f(c k ; c l ) is non-zero at fc k = ; c l = g or fc k = ; c l = g ut 2 f(c k ; c l ) is just c k c l c k c l det( ^V l ), proving the result We now in a position to present and prove the main result Theorem 34 The maximum order of an s-stage MIRK cannot exceed s + Proof: If s = or 2, then the result follows from sections 2 and 22 Hence we assume s 3 Suppose there is an s-stage MIRK of order s + 2 We consider the T's one at a time T() states that (c 2? c )y ; =, and hence at least one of these two factors must vanish Suppose c 2? c = Then, in fact, y ; must also be zero, for consider T (2), which becomes! c 2 2? c 2 y 2;2 =? c 3 2? c 2?! If y ; 6=, then c 2 = or, both of which are impossible Hence y ; = y = This result holds independently of whether or not c 2 = c, c 2 2 = c 2, or c = c 2 Note that ky y ; or det(v k ) = ( i=(c 2 i? c i ))( Y (c i? c j )); i<j and hence if all the c i are distinct and dierent from zero or one, then the T's from () through (s? ) give y i = for all i = ; : : :; s? : T(s) then obviously fails to hold Thus we see that if there is to be any possibility for an order s+2 method to exist, we must satisfy at least one of the conditions 2! y ; : c 2 i = c i ; i 2 f; : : :; sg or c i = c j for i 6= j; i; j 2 f; : : : ; sg: 6 ;
We shall refer to these as singularities of type I (c 2 i = c i ) and type II (c i = c j ; i 6= j) Suppose the rst singularity that we encounter is of type II and involves c k, ie c k = c i for one i k? lso, since this is the rst singularity, det(v k? ) 6= ; and y i = ; i = : : : k? : T(k) becomes c 2? c : : : c 2 k?? c k? : : : c k+? c : : : c k+ k?? c k? y k; y k;i + y k;k = : y k;k? We see the eect of a singularity of this type is to replace the y k;i term in y k by y k;i + y k;k, and reduce the dimension of this T by one Furthermore this same eect occurs in all the subsequent T's, (k + ); : : :; s Indeed, all subsequent type II singularities produce this same kind of eect Each singularity leads to a modication of the y i vectors and the T's, with a reduction by one in the size of each set of conditions Thus if we modify the original T's (and y i vectors) according to the presence of all type II singularities, the result will be a set of modied T's, smaller in dimension than the original set but with the same structure The advantage of considering this modied set of T's is that we can restrict our attention to only type I singularities, ie we can assume that all abscissa are distinct In the remainder of this proof we will use the same notation as above for the T's and y i vectors but will assume that the system of conditions has been reduced to remove all type II singularities Let us now consider this set of reduced T's, and suppose that no type I singularities are present Then, T() gives y =, and in each subsequent set of conditions, V k ; k = 2; : : :; s? is nonsingular, so we get, y i = The nal condition, T(s), then leads to an obvious contradiction (This contradiction is obtained no matter what the dimension of reduced T's) Hence, there must be at least one type I singularity if there is to be any chance for the existence of an s-stage, order s + 2 MIRK scheme 7
(In fact, there can be at most two type I singularities The rst type I singularity would imply that the corresponding abscissa is or The second would similarly imply that the corresponding abscissa is or, but dierent than that of the rst, since we have assumed the absence of all type II singularities The third type I singularity would imply that the corresponding abscissa is or, and thus equal to one of the previous two abscissas associated with the type I singularities, an impossibility, given the absence of all type II singularities) We now continue our examination of the reduced T's and assume that the rst type I singularity involves c k Then c 2 k = c k and T(k) gives c 2? c : : : c 2 k?? c k? : : : c k+? c : : : c k+ k?? c k? y k; y k;k? = ; and since this coecient matrix is nonsingular, y k; = : : : = y k;k? = and y k;k is arbitrary T(k + ) now gives c 2? c : : : c 2 k?? c k? 2c k? c 2 k+? c k+ c k+2? c : : : c k+2 k?? c k? (k + 2)c k+ k? c k+2 k+? c k+ y k+; y k+;k? y k;k y k+;k+ = : learly y k+;k is arbitrary and its place has been taken by y k;k Suppose that there is exactly one type I singularity present Then the above coecient matrix is seen to be nonsingular by interchanging the last two columns and using lemma 32, and thus we have y k;k =, y k+;k arbitrary, and all the remaining components of y k+ equal to zero 8
If we next consider T(k + 2), we get a similar structure, c 2? c : : : c 2 k?? c k? 2c k? c 2 k+? c k+ c 2 k+2? c k+2 c k+3? c : : : c k+3 k?? c k? (k + 3)c k+2 k? c k+3 k+? c k+ c k+3 k+2? c k+2 gain, lemma 32 gives y k+;k = and all components of y k+2 y k+2; y k+2;k? y k+;k y k+2;k+ y k+2;k+2 = : equal to zero except the kth one, which is arbitrary This same eect obviously continues for each subsequent T and corresponding y i vector, leading to the observation that y s? has all zero exponents, except for the kth one, which is arbitrary Then T(s) becomes, 2c k? y s?;k =? 2 (s + )c s? k where c k = or learly the rst two equations of the above condition lead to a contradiction Hence, if there is any chance for an s stage MIRK scheme to be of order s + 2, there must be two type I singularities (Note that if all the abscissa are equal, then there can be at most one type I singularity, and the proof ends here) The nal possibility is that there is a second type I singularity, which we will suppose involves c l ; l > k That is c 2 l 3 2 4 s s+2 = c l with c l 6= c k Then T(l) has the form, c 2? c : : : c 2 k?? c k? 2c k? c 2 k+? c k+ : : : c 2 l?? c l? c l+? c : : : c l+ k?? c k? (l + )c l k? cl+ k+? c k+ : : : c l+ l?? c l? ; y l; y l;k? y l?;k y l;k+ with y l;k and y l;l, arbitrary Rearranging columns of the above coecient matrix and applying lemma 32 shows that the above matrix is nonsingular and thus y l?;k =, and all components of y l are zero, except for the kth and lth, which are arbitrary The (l + )st T becomes, 9 y l;l? = ;
c 2? c : : : c 2 k?? c k? 2c k? c 2 k+? c k+ : : : c 2 l?? c l? 2c l? c l+2? c : : : c l+2 k?? c k? (l + 2)c l+ k? c l+2 k+? c k+ : : : c l+2 l?? c l? (l + 2)c l+ l? y l+; y l+;k? y l;k y l+;k+ y l+;l? = ; y l;l with y l+;k and y l+;l arbitrary Lemma 33 shows that the above coecient matrix is nonsingular and thus we have y l;k = y l;l =, and all components of y l+ equal to zero, except the kth and lth, which are arbitrary Obviously this eect continues through the subsequent T's, and we reach the conclusion that y s? has all components equal to zero, except the kth and lth, which are arbitrary T(s) becomes 2c k? y s?;k + (s + )c s? k 2c l? y s?;l =? 2 (s + )c s? l with c k = and c l =, or c k = and c l = The rst three equations in this system are then impossible to satisfy This completes the nal possibility for the existence of an s-stage, order s + 2 MIRK scheme, and we conclude that no such scheme can exist 4 onclusions In this paper we have extended the knowledge of the class of mono-implicit Runge-Kutta methods in several ways We have completely characterized many of the lower stage, lower order families of this class which have optimal stage order This is signicant, when solving sti problems for example, because it is now known that in this case the stage order rather than the classical order is the appropriate measure of the accuracy of the method (see [5]) These characterizations will be useful in an analysis for determination of new formulas for use in a boundary value ODE code using defect control, under development by one of the authors general result giving a maximum bound of 3 for the stage order of an s-stage MIRK formula was given main result of this paper shows that the order of an s-stage MIRK 2 3 2 4 s s+2 ;
method is bounded by s + >From the results of section 2, we have presented formulas having s = ; 2; 3; 4, and 5, for which this bound is met We conjecture that the bound cannot be met for s 6 Future work in this area could include a systematic investigation of the linear and nonlinear stability properties of the various families of MIRK methods identied in section 2, as well as possible further attempts to completely characterize low stage MIRK methods having lower stage orders lso, an investigation of the possibility of embedded families of MIRK methods of various orders would be very useful for error estimation purposes in software implementations for both in the initial value and boundary value problem areas (Some embedded families of MIRK methods have been presented in [] and [6]) 5 References [] W M G van okhoven, Ecient higher order implicit one-step methods for integration of sti dierential equations, IT 2 (98), 34{43 [2] K urrage, special family of Runge-Kutta methods for solving sti dierential equations, IT 8 (978), 22{4 [3] K urrage, Order properties of implicit multivalue methods for ordinary dierential equations, IM J Num nal 8 (989), 43{69 [4] J utcher, Implicit Runge-Kutta processes, Math omp 8 (964), 5{64 [5] J utcher, \The Numerical nalysis of Ordinary Dierential Equations," John Wiley & Sons, Toronto, 987 [6] J R ash, class of implicit Runge-Kutta methods for the numerical integration of sti dierential systems, J M 22 (975), 54{5 [7] J R ash, On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections, Part : survey and comparison of some one-step formulae, omput Math ppl 2a (986), 29{48 [8] J R ash, On the numerical integration of nonlinear two-point boundary value problems using iterated deferred corrections, Part 2: The development and analysis of highly stable 2
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