On Implicit Runge-Kutta Methods with a Stability Function Having Distinct Real Poles

Transcription

1 On Implicit Runge-Kutta Methods with a Stability Function Having Distinct Real Poles Stephen L Keeling * Abstract Because of their potential for offering a computational speed-up when used on certain multiprocessor computers, implicit Runge-Kutta methods with a stability function having distinct poles are analyzed These are called multiply implicit (MIRK) methods, and because of the so-called order reduction phenomenon, their poles are required to be real, ie, only real MIRK s are considered Specifically, it is proved that a necessary condition for a q-stage, real MIRK to be A-stable with maximal order q + is that q =,, 3, or 5 Nevertheless, it is shown that for every positive integer q, there exists a q-stage, real MIRK which is strongly A -stable with order q +, and for every even q, there is a q-stage, real MIRK which is I-stable with order q Finally, some useful examples of algebraically stable real MIRK s are given Key words: Implicit Runge-Kutta methods, parallel computations, order reduction, real-pole sandwich 98 Mathematics Subject Classification: 65M99 Introduction This paper is concerned with the characterization and construction of implicit Runge-Kutta methods (IRKM s) with a stability function having distinct real poles The work is motivated by the potential that such methods offer a computational speed-up when used on parallel machines with a modest number of processors For a precise discussion of the issues, IRKM s and their properties are now introduced Given an integer q, a q-stage IRKM is determined by a set of constants: a a q τ a q a qq τ q b b q and it is convenient to make the following definitions: A {a ij } q i,j=, bt b,b,,b q, B diag i q {b i}, M BA + A T B bb T, T diag i q {τ i}, V {τ j i } q i,j=, R diag i q {i }, e T,,, For the IRKM formulation used in this work, choose arbitrarily, t R, y R n, F : R n+ R n sufficiently smooth, and k > sufficiently small, so that for t t t +k, smooth functions y,ŷ : R R n are well-defined by: { Dt y(t) = F(t,y(t)) y(t ) = y, () * Department of Mathematics, Vanderbilt University, Nashville, TN 3735 This work was supported by the National Aeronautics and Space Administration under NASA Contract No NAS-87 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA

2 and: y i (t) = y + (t t ) a ij F(t + τ j (t t ),y j (t)), j q j= () ŷ(t) = y + (t t ) b i F(t + τ i (t t ),y i (t)) i= The method is described as explicit if a ij =, i j and implicit if for any i, a ii As usual, there are three criteria by which a method is judged in this work: order of consistency, stability, and implementability First, an IRKM is said to have order ν if for every y and ŷ defined as above, Dty(t l ) = D tŷ(t l ), l ν Nevertheless, when these methods are used for stiff problems, they suffer from an order reduction phenomenon Specifically, if p is the largest integer for which the following holds: T[e;Ae; ;A p e] = [Ae;A e; ;(p )A p e] then it often happens that only a k min(ν,p) type convergence can be proved or demonstrated computationally (See eg, [6], [7], [4], [], [3], [6], and []) Also, note the barrier: p q+ With regard to stability, let r(z) be a rational approximation to the exponential e z defined by: r(z) zb T (I + za) e (3) This is the so-called stability function whose poles correspond to the negative reciprocals of the eigenvalues of A The stability function is used to define certain notions of stability as follows An IRKM is said to be A -stable if: r(z) z, and strongly A -stable if: It is I-stable if: A-stable if: and algebraically stable if: sup z z r(z) < z > r(z) R{z} =, r(z) R{z}, B and M are positive semidefinite Note that the latter is stronger than A-stability [7] In fact, for the methods which are strongly A -stable and algebraically stable with B actually positive, there exist results for certain parabolic equations which guarantee decay of approximations with respect to the time step [] Now, concerning implementability, there has been much effort devoted to the development of IRKM s for which the poles of r(z) are identical and real ([8], [9]) This is due to the well-known fact that these so-called singly implicit (SIRK) methods offer a computational advantage over other IRKM s on serial machines However, it is noted in [] - [3] that in a parallel environment the preferred methods may actually be those for which the poles of r(z) are distinct In this work, the latter are referred to as multiply implicit (MIRK) methods Further, they are called real if σ(a) R, and otherwise complex It is explained in [3] that real MIRK s permit a greater degree of parallelism than those which are complex However, if the poles of r(z) are real, then ν q + [5] Nevertheless, while the classical order can be increased at the cost of introducing complex poles, the order

3 reduction phenomenon enforces the p q + barrier for stiff problems Hence, the principal interest here is in real MIRK s In this connection, it is important for certain parabolic problems that for every positive integer q, there exists a q-stage, real MIRK which is (strongly) A -stable with maximal order ν = q + and p = q + This fact follows from results in [], but an independent and direct proof is given in section 3 Also, for hyperbolic problems, it is shown in section 4 that for every even integer q, there is a q-stage, real MIRK which is I-stable with order ν = q For a related result, see Bales, Karakashian, and Serbin [] Concerning methods which are more stable, Wanner, Hairer, and Nørsett [7] have established that if an A-stable SIRK has order ν = q +, then q =,, 3, or 5 Also, for SIRK s which are actually algebraically stable, the limit p = q + is achievable only for q [3] Nevertheless, it is shown in [3], that for a wide range of problems, an IRKM can be modified in such a way that the order reduction is no worse than k min(ν,q+) even if p < q + Hence, without regard for the value of p, one is led to ask about the existence of real MIRK s which are A-stable with maximal order ν = q + In section, it is shown that a necessary condition on such methods is that q =,, 3, or 5, ie, the result of [7] is generalized to the case of real, distinct poles Then in section 5, some useful examples of algebraically stable real MIRK s are presented A Barrier for A-Stable Real MIRK s with Maximal Order In this section, it is proved that a necessary condition for a q-stage, real MIRK to be A- stable with maximal order ν = q +, is that q =,, 3, or 5 The plan of the proof is roughly as follows According to the work of Nørsett and Wanner [4], for a real MIRK to be A-stable with maximal order, the poles of the rational function (3) must be linked to a certain type of hypersurface near the middle of the so-called real-pole sandwich However, it is shown below that when constrained to such hypersurfaces, r( ) only if q =,, 3, or 5 The latter claim is established by considering r( ) as a function depending on the poles of r(z), and showing that it is minimized over the positive part of a maximal order hypersurface of the real-pole sandwich at the point where the poles are the same Then since such points have been studied carefully for SIRK s, the remainder of the proof follows from the work of Wanner, Hairer, and Nørsett [7] Before proceeding with the details, the appropriate notation is developed First, define the so-called symmetric polynomials for x R m : S (x), S l (x) i <i < <i l m x i x i x il l m, S m+ (x) Next, some relevant properties of these polynomials are enumerated By direct calculation: Thus: S l (x) = x i i < <i l m i j i, j l x i x il + i < <i l m i j i, j l x i x il = x i xi S l (x) + xi S l+ (x) i m, l m x R m () xi S l+ (x) = S l (x) xi = i m, l m x R m () Also with e m R m having all unit coordinates: S l (xe m ) = x l i <i < <i l m = x l ( m l ) (3) 3

4 since the sum is the number of combinations of m distinct integers taken l at a time Finally with x x,,x m T :,x S l (x) S m (x) = S m l(x ) l m x R m (4) Now according to [4] and (4), the rational function (3) has real poles and order of approximation to e z equal to at least q, * if and only if it has the form: r(z) = ( z) l l m= ( ) m (l m)! S m(x ) S l (x )z l = l ( z) l ( ) m (l m)! S q m(x) m= (5) S q l (x)z l for some x R q Also since: q S q l (x)z l = (x m + z) m= the poles of r(z) are { x m } q m= So it follows from work in [4] that a q-stage, real MIRK can be A-stable with maximal order q + only if x is contained in one of the q hypersurfaces of the set: W {x R q ( ) l : x i >, (l + )! S l(x) = } For convenience, these sheets or connected components of W are ordered as follows First, define the Laguerre polynomial of degree m: ( m L m (x) ( ) l m l ) x l l! and the values {x m i }m i m by: Then using (3): L m+(x m i ) =, < x m < x m < < x m m W W W W q with ex q i W i, i q Now, the following is an adaptation of Theorem of [4] Lemma Let an A-stable, q-stage IRKM be given with a rational function (3) having only real poles and maximal order q + Then (3) must have the form (5) and it is necessary that: where W x W q+ W q if q is odd, or The next lemma is proved in section 5 of [7] x W q if q is even * This notion of order is of course weaker than that described in the Introduction for IRKM s, but the distinction should be clear from the context The maximal order hypersurfaces of the real-pole sandwich are defined in [4] as the sheets of M {y R q : q ( ) l S (l+)! q l(y) = }; so by (4), to every point in W there corresponds a point in M with reciprocal coordinates 4

5 Lemma For m, i, L m (x m i ) only if:, i = 5, i = m 9, i = 3 6i, i 4 Recalling (5), define: R(x) ( ) l S l (x) = r( ) l! Next, let an IRKM be given as specified in Lemma, so that in particular, r( ) Now suppose that the following has been established: inf R(x) = L q (x q x W i ) i q (6) i Then according to Lemma, one of the following must hold: q is odd, x W q+, and: L q (x q q+) q is odd, x W q, and: L q (x q q ) q is even x W q, and: L q (x q q ) Finally according to Lemma, the first case implies that q =, the second that q = 3 or 5, and the third that q = This is the advertised result and what remains is to establish (6) Lemma 3 The constrained critical points of R(x) on W are precisely those points in W with q identical coordinates Proof: First, form the Lagrangian: ( ) l ( ) l F(x,λ) S l (x) λ l! (l + )! S l(x) so that if R(x) has a constrained critical point at x W, then: q = xi F(x,λ) = Also by (): ( ) l q (l + )! x i S l+ (x ) + λ R(x) ( ) l (l + )! x i S l+ (x ) i q (7) = ( ) l (l + )! S l(x ) = x i ( ) l (l + )! x i S l (x ) + l= q ( ) l (l + )! x i S l+ (x ) = x i q ( ) l (l + )! x i S l+ (x ) + q ( ) l (l + )! x i S l+ (x ) (8) i q Now for the sake of contradiction suppose that for some j: q ( ) l (l + )! x j S l+ (x ) = (9) 5

6 Then it follows from (8) that in addition: q ( ) l (l + )! x j S l+ (x ) = () Using () and (4), it follows that if the coordinates of y R q are reciprocal to those of x other that x j, then: xj S l+ (x ) xj S q (x ) = S l(y ) S q (y ) = S q l(y ) So, dividing (9) and () by xj S q (x ) gives the following conditions: q ( ) l (l + )! S q l(y ) = = q ( ) l (l + )! S q l(y ) However, as indicated in [4], with: n N (t;y) ( ) n l S n l (y) tl t l!, N m+(t;y) N m (s;y)ds the following must be satisfied: y R n () {y R n : N (;y) = } {y R n : N (;y) = } = Hence, (9) and () cannot hold simultaneously Therefore, by (7) and (8), x i = λ = x j, i,j q Now in order to capture the global minima of R(x) on the components of W, the following corollary is concerned with constrained critical points on the boundary of W Its proof involves only appropriate adjustments in the dimensions of the above argument Corollary Given a multi-index m m,m,,m q where m i = or, i q, and m q i= m i, m q, define: R m + {x R q : x i =, if m i =, x j >, if m j =, i,j q} Then the constrained critical points of R(x) on W R m + are precisely those points in W R m + with m identical coordinates The next lemma is not actually contained in Theorem 8 of [4], but the argument required is essentially the same Nevertheless, a brief proof is offered for completeness Lemma 4 Let m = m, m q, and suppose e m R m + has only unit nonzero coordinates Then the points of W R m + with m identical coordinates are precisely: x m i em W i i m, and: R(x m i em ) = L m (x m i ) i m Proof: Fix m, m = m, with m q, and n, n = n, with n = m Also, suppose R n + Rm + and that em R m + and en R n + have only unit nonzero coordinates Next, with X() {x R m + : x = xem,x > } and X() {x R n + : x = xen,x > }, suppose that for θ [,), X(θ) is a ray from the origin into R m + passing smoothly from X() to X() With (3), it follows that X() W = {x m i em } m i= and X() W = {xn i en } n i= So for θ [,), 6

7 let {x i (θ)} m i= denote the points of X(θ) W ordered according to increasing magnitude That these points remain separated, preserved in number and order, and bounded for θ in compact subintervals of [, ), can be seen by noting that the explicit calculation of their coordinates involves the roots of a polynomial which has degree m while θ < Since the components of W never intersect, [4] these roots can never coalesce to become multiple or complex or to change relative positions Now, since their number decreases by one as θ passes to, it only remains to determine whether x m escapes to or x passes through the origin Since W =, the indicated alignment between the points {x m i em } m q i m and {W i} i q is established Finally, the values for R(x) are obtained with (3) Now the following shows that the constrained critical values of R(x) determine the constrained global minima of R(x) Lemma 5 The constrained global minima of R(x) on the components of W are given as follows: inf R(x) = min { L m(x m i ) } i q x W i i m q Proof: First, note that R(x) never vanishes on W, because the rational function (5) cannot have degree q in the numerator and order q + simultaneously [5] Hence, on each W i R m +, R(x) is a smooth, positive polynomial with exactly one critical point Since R(x) is a polynomial, R(x) as x Therefore, for a given i, R(x) must be minimized over W i R m + at its critical point The rest of the result follows with Corollary and Lemma 4 Now, the following lemma yields (6) Lemma 6 The Laguerre polynomials {L m (x)} m, and the values {x m i }m i m satisfy: Proof: By the identity: L m (x m i ) < L m (x m i ) i m, m () L m (x) = L m+ (x) the inequalities of () are equivalent to: x m + L m+ (x) (3) L m+ (x m i ) < L m(x m i ) i m, m which can be established by showing that: and: L m (x m i ) < L m (x m i ) = L m+(x m i ) < for odd i m (4) < L m+ (x m i ) = L m (x m i ) < L m (x m i ) for even i m (5) First note that since L m+ (x) has exactly m+ simple real roots, it has exactly m local extrema which must be associated with the m simple real roots of L m+ (x) Therefore since L m+() = : So using this with (3): ( ) i L m+ (x m i ) >, ( )i+ L m+ (xm i ) > i m (6) ( ) i L m(x m i ) = ( )i+ x m i m + x m i L m+(x m i ) = m + L m+(x m i ) > i m Now assume that i m is odd Then on the interval [x m i,xm i+ ], L m(x) is decreasing at the beginning and increasing at the end, while L m(x) vanishes only at xi m (x m i,xm i+ ) where L m (x) is minimized Combining this fact with (3) and (6) gives (4) Also (5) follows similarly Finally, the result of this section is summarized in the following 7

8 Theorem Let an A-stable, q-stage IRKM be given with a rational function (3) having only real poles and maximal order q + Then q =,, 3, or 5 3 A Family of High Order Strongly A -Stable Real MIRK s In spite of the barrier established above, a family of IRKM s is constructed in this section, which contains for every positive integer q, a q-stage, real MIRK which is strongly A -stable with maximal order q + The importance of such a family can be seen as follows Let S h be a finite dimensional space and suppose that an approximation is required for the solution u h : [,t ] S h, to the initial value problem: { Dt u h = L h (t)u h + f(t,u h ) t t u h () = u h where h amounts to a stiffness parameter, and for simplicity, f is assumed to be smooth and independent of h Also L h is assumed to be linear and selfadjoint with positive spectrum Such a problem could of course arise from the semidiscretization of a semilinear parabolic initial boundary value problem Further, A -stable methods are useful for for parabolic equations, since the following is pivotal in the stability analysis: r(kl n h ) where is an appropriate norm [] Note that if L h is selfadjoint and positive definite, then this inequality follows from A -stability and a spectral argument Otherwise, a restrictive relationship between h and k must be imposed Toward the goal of this section, recall the functions {N m (t;y)} m defined in () Then let ê e,e,,e q T be a unit vector chosen so that: e i > and e i e j i j q According to [4], there are exactly q positive solutions to: N (,yê) = So if y is the largest, set y y ê Then it follows that the function: G(t) N (t,y ) N (,y ) = tq+ N (,t y ) ( ) q S q (y ) vanishes at t = and t = but for no t (,) Further, since G () =, it follows that for every t [,], G (t) < Finally, note that by (4): G () = ( ) l l! S q l (y ) S q (y ) = R(y ) Lemma 3 G () < Proof: As explained in the proof of Lemma 5, R(x) never vanishes on W Also, from the proof of Lemma 4, it can be seen that W is compact So, the global maximum of R(x) on W is determined by the constrained critical values of R(x) given by Corollary and Lemma 4 Hence by Lemma 6: sup x W R(x) = max m q { L m(x m ) } = L (x ) 8

9 Then since some calculations show that y W, G () L (x ) = Now set: l ( z) l ( ) m (l m)! S m(y ) m= R(z) S l (y )z l taking Q(z) as the denominator Theorem 3 Any IRKM for which (3) is equal to R(z) above, must be A -stable Proof: First, note the error formula of [4]: R(z) = e z [ With x, integration by parts gives: or: where: e x R(x) = + ( x)q Q(x) = + xq S q (y ) Q(x) ( z)q+ Q(z) e zt N (t;y )dt [ ] e x N (;y ) N (;y ) e xt N (t;y )dt [ ] (e x ) + (e x e xt )G (t)dt R(x) = e x ( θ(x)) + θ(x)( + J(x)) θ(x) xq S q (y ) [,), and J(x) S l (y )x l As indicated above, G (t) < and G () <, so: Hence: ] ( e x( t) )G (t)dt J(x) = J(x) G (t)dt = G () < R(x) = R(x) = e x ( θ(x)) + θ(x)( + J(x)) = x =, < < x < x Now the next three theorems show that R(z) can be used to construct a q-stage, real MIRK which is strongly A -stable with order q + The first is due to Bales, Karakashian, and Serbin [] Theorem 3 If the coordinates of y R q are distinct and nonzero, then N (t;y ) has q distinct real roots {τ i } q i= Theorem 33 Given distinct real roots {τ i } q i= of N (t;y ), the following are well-defined: A TV RV and b (V T ) Re and the resulting IRKM has order at least q Furthermore, if N (,y ) =, then the order is q + 9

10 Proof: With the conditions of Butcher: B(ξ) : lb T T l e = l ξ, C(ξ) : lat l e = T l e l ξ, conditions B(ν) and C(µ) imply order ν if ν µ + [5] Now the first statement follows since the assumptions amount to conditions C(q) and B(q) Then if N (,y ) =, by B(q): q + = q t q dt = q! [N (t;y ) ( ) q l S q l (y ) tl l! ]dt q = q!n (,y ( ) l ) q! S q l (y ) b i τi l l! i= = b i [τ q i q!n (τ i,y )] = b T T q e i= and the second statement follows with B(q + ) and C(q) Finally, the next theorem follows from Theorems and 4 of [4] Theorem 34 Given an IRKM constructed as described in Theorem 33, the rational function (3) must have the form (5) with x = y 4 A Family of High Order I-Stable Real MIRK s In this section, a family of IRKM s is constructed which contains for every even q, a q-stage, real MIRK which is I-stable with order q As briefly indicated below, such methods are useful for hyperbolic problems Suppose that an approximation is required for the solution u h : [,t ] S h, to the initial value problem: Dt u h = L h (t)u h + f(t,u h ) t t u h () = u h D t u h () = u h where L h and f are assumed to have the same properties as mentioned in section 3 Note that this problem can be expressed in first order form as follows: ( ) ( ) ( ) ( ) uh I uh D t = + t t D t u h L h (t) D t u h f(t,u h ) ( ) ( ) uh () u = h D t u h () u h As opposed to the parabolic case, with: L n h ( I L n h ) the pivotal stability inequality here is: r(kl n h) where is an appropriate norm Since L n h has only imaginary eigenvalues, a certain spectral argument shows that I-stability is crucial []

11 Now in the sequel, q is implicitly assumed to be even Toward the goal of this section, recall the functions {N m (t;y)} m defined in () Then let ê e,e,,e q T be a unit vector chosen so that: e i = e q i+ > and e i e j i j q According to [4], there are q real solutions to: N (,yê) = and exactly q of them are positive So if y is the largest, choose y y and set y y ê Then it follows that: G(t) N (t,y ) N (,y ) = tq N (,t y ) ( ) q S q (y ) decreases in a strictly monotonic fashion from to G() as t passes from to Now set: Q(z) q m= [ (y mz) ] = S l (y )z l, P(z) ( z) l l m= ( ) m (l m)! S m(y ) and take R(z) P(z)/Q(z) Theorem 4 Any IRKM for which (3) is equal to R(z) above, must be I-stable Proof: First, recall the error formula of [4]: So with z = iy: Integration by parts gives: where: θ(y) R(z) = e z [ ( z)q+ Q(z) e zt N (t;y )dt q R(iy) = iy (yy m ) [ + (yy m= m) e iyt G(t)dt ] R(iy) = ( θ(y)) + θ(y)(e iy G() J(y)) q m= By construction, G (t) <, so: Further, since G() : (yy m ) [ + (yym ) ] [,) and J(y) e iyt G (t)dt J(y) G (t)dt = G() R(z) ( θ(y)) + θ(y)g() + θ(y)( G()) ] Now, in view of Theorems 3-34, for every even integer q, the rational function R(z) above can be used to construct a q-stage, real MIRK which is I-stable with order q

12 5 Some Algebraically Stable Real MIRK s Since algebraic stability implies A-stability, the results of section imply that there can exist a q-stage real MIRK which is algebraically stable with maximal order q +, only if q =,, 3, or 5 In this section, such methods are actually constructed for the cases q = and 3 Note that this is also possible for the case q = 5, but the details are not given here In the discussion, use is made of Theorem 5 below, which is due to Hairer and Wanner [9] For this, some relevant constructions are now provided Given that B >, let C be determined by the Cholesky decomposition: V T BV C T C Then take: so that: Finally with: δ i m W V C W T BW = I { i m otherwise let I m {δ ij δ i m } q i,j=, H {(i + j ) } q i,j=, and: X G diag q q {,,,,} + subdiag {ξ i} supdiag {ξ i}, i q i q Now the following can be stated ξ i 4i Theorem 5 Let an IRKM be given for which B > and: I i V T BV I j = I i HI j i + j = ν Let m [ (ν )] and suppose X is a q q matrix satisfying: XI m = X G I m, I m X = I m X G and if ν is even: Then if A = WXW the IRKM has order ν ê T m+xê m+ = First, the following family of two-stage methods has been described by Karakashian []: τ (τ τ ) τ τ τ τ τ τ τ (λ + λ ) (λ + λ ) τ τ τ τ ( τ τ ) τ τ τ τ (λ + λ ) + (λ + λ ) τ τ τ The W-transformation leads to: 3(τ ) W = 3(τ ) τ λ > τ τ, λ = λ λ and X = 3 3 x 6

13 where: x λ + λ > Since B > and V T BV = H, by Theorem 5, these methods have maximal order ν = 3 With regard to stability, since W T BW = I, the algebraic stability matrix M satisfies: W T MW = W T BWW AW + (W AW) T W T BW + W T BW[W e(w e) T ]W T BW = X + X T ê ê T = diag{,x} Hence, these methods are algebraically stable In fact, r( ) = 6 λ λ < So, since algebraic stability implies A-stability, strong A -stability follows with the maximum principle Also note that r(z) has distinct real poles since A has real, distinct eigenvalues: A = S ΛS, Λ = diag i {λ i}, S = [ τ λ τ λ τ λ τ λ Next, the following family of three-stage methods has been described in [3] First, choose λ, λ, and λ 3 distinctly and to satisfy: λ,λ,λ 3 > λ + λ + λ 3 > and λ 3 = For example, it is sufficient to choose λ 3 as indicated after taking: λ λ 6 (λ + λ ) + 4 λ λ (λ + λ ) + 6 ] λ > and λ > λ 6 λ Now define: x λ + λ + λ 3, x [(λ λ + λ λ 3 + λ λ 3 ) (λ + λ + λ 3 ) + 6 ], and: Also take Λ = diag i 3 {λ i}, X 3 α i λ i x, β i λ i i 3 3 α α α 3 x, and Y 3α β 3α β 3α 3 β 3 x x 3x β 3x β 3x β 3 Then X = Y ΛY Now choose τ, τ, and τ 3 to satisfy: (τ τ + 6 )(τ τ + 6 ) τ τ (τ + τ ) + 3 < and τ 3 = τ τ 3 (τ + τ ) + 4 τ τ (τ + τ ) + 3 For example, it is sufficient to choose τ 3 as indicated after taking: Then it can be shown that: τ < ( 3 ) and ( + ) < τ 3 σ τ τ τ 3 3 (τ τ + τ τ 3 + τ τ 3 ) + 4 (τ + τ + τ 3 ) >

14 so take r (σ 7 36 ) and define W by: 3(τ ) (τ τ + 6 )/r W 3(τ ) (τ τ + 6 )/r 3(τ 3 ) (τ 3 τ )/r Next, since it can be shown that τ i τ j, i j, choose the vector b according to: Finally take: b = (V T ) Re A WXW = S ΛS, S Y W and note that r(z) has distinct real poles since A has real, distinct eigenvalues Concerning stability, note that B > since V T BV = H + diag{,,σ 5 } Also since W T BW = I, the algebraic stability matrix M satisfies: W T MW = W T BWW AW + (W AW) T W T BW + W T BW[W e(w e) T ]W T BW = X + X T ê ê T = diag{,,x } Hence these methods are algebraically stable Further, with max{λ,λ,λ 3 } >, it follows from the work in [] that these methods are strongly A -stable Now since B > and V T BV is as indicated above, it follows from Theorem 5 that these methods have maximal order ν = 4 References [] Bales, L A, Karakashian, O A, Serbin, S M, On the A Stability of Rational Approximations to the Exponential Function with Only Real Poles, BIT, v 8, 988, pp 7-79 [] Bales, L A, Karakashian, O A, Serbin, S M, On the Stability of Rational Approximations to the Cosine with Only Real Poles (To appear in BIT) [3] Burrage, K, High Order Algebraically Stable Runge-Kutta Methods, BIT, v 8, 978, pp [4] Burrage, K, Hundsdorfer, W H, Verwer, J G, A Study of B-Convergence of Runge-Kutta Methods, Computing, v 36, 986, pp 7-34 [5] Butcher, J C, Implicit Runge-Kutta Processes, Math Comp, 8, 964, pp 5-64 [6] Crouzeix, M, Sur l approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta, Thèse, Université de Paris VI, 975 [7] Dekker, K, Verwer, J G, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, New York, Oxford, 984 [8] Hairer, E, Highest Possible Order of Algebraically Stable Diagonally Implicit Runge- Kutta Methods, BIT, v, 98, pp [9] Hairer, E, Wanner, G, Algebraically Stable and Implementable Runge-Kutta Methods of High Order, SIAM J Numer Anal, v 8, 98, pp

15 [] Karakashian, O A, On Runge-Kutta Methods for Parabolic Problems with Time Dependent Coefficients, Math Comp, v 47, 986, pp 77- [] Karakashian, O A, On Runge-Kutta-Nyström Methods for Hyperbolic Problems with Time Dependent Coefficients, Comput Math Applic, v 3, 987, pp [] Karakashian, O A, Rust, W, On the Parallel Implementation of Implicit Runge- Kutta Methods (To appear in SIAM J Sci Stat Comput) [3] Keeling, S L, Galerkin/Runge-Kutta Discretizations for Parabolic Partial Differential Equations, PhD Dissertation, University of Tennessee, 986 [4] Nørsett, S P, Wanner, G, The Real-Pole Sandwich for Rational Approximations and Oscillation Equations, BIT, v 9, 979, pp [5] Nørsett, S P, Wolfbrandt, Attainable Order of Rational Approximations to the Exponential Function with Only Real Poles, BIT, v 7, 977, pp -8 [6] Sanz-Serna, J G, Verwer, J G, Hundsdorfer, W H, Convergence and Order Reduction of Runge-Kutta Schemes Applied to Evolutionary Problems in Partial Differential Equations, Numer Math v 5, 987, pp [7] Wanner, G, Hairer, E, Nørsett, S P, Order Stars and Stability Theorems, BIT, v 8, 978, pp