CLASSIFICATION OF HIGH-ORDER IMPLICIT RUNGE-KUTTA … · 2014-06-12 · NEW ZEALAND JOURNAL OF MATHEMATICS Volume 29 (2000), 151-167 CLASSIFICATION OF HIGH-ORDER IMPLICIT RUNGE-KUTTA

NEW ZEALAND JOURNAL OF MATHEMATICS Volume 29 (2000), 151-167

CLASSIFICATION OF HIGH-ORDER IMPLICIT RUNGE-KUTTA METHODS

Abstract. This paper presents a characterization of the properties of implicit Runge-Kutta methods of high order using the ^-transform ation of Hairer and Wanner [8]. This approach enables the conditions for symplecticity, symmetry, algebraic stability and stiff accuracy to be expressed simply and allows classes of methods that possess some combination of these attributes to be constructed in a natural way. We consider s-stage methods of order greater than or equal to 2s — 2 with abscissas chosen as the real and distinct zeros of the two-parameter family, Ps {2x — 1) + rPs- i ( 2 x — 1) + tPs-2 (2 x — 1), t < (s — l ) /s , where the Pfc(2x — 1) are the Legendre polynomials orthogonal on [0,1]. In particular, a simple characterization of stiff accuracy for these methods and conditions for their stability functions to be Pade approximations are given. The inter-relationships of the various properties examined in this general setting therefore includes those for methods based on well-known quadrature formulas as well as their generalizations.

1. Introduction

Implicit Runge-Kutta methods were originally proposed for the numerical solution of stiff ordinary differential equations because of their stability properties. Methods which possess linear and nonlinear stability properties have been constructed and studied extensively. More recently, implicit Runge-Kutta methods have been proposed and applied to differential-algebraic equations, and Hamiltonian problems with and without constraints. The properties of stiff accuracy or symplecticity are important for these newer applications. Thus in searching for methods with a combination of these various attributes it will be convenient to have a systematic approach based on their simple characterizations. This paper presents such a general approach to the construction of s-stage methods of order greater than or equal to 2s — 2.

We consider the polynomial M of degree s defined by

1991 A M S Mathematics Subject Classification: 65L05, 65L06.K ey words and phrases: symplecticity, symmetry, stiff accuracy, algebraic stability, A —stability, Pade approximations, partitioned Runge-Kutta methods.*The work of this author was supported (in part) by IRISA.

R.P.K . C h a n * a n d P. C h a r t i e r

(Received February 1999)

M {x ) = Ps(x) +s — 1

s r e\R, (1)

152 R.P.K. CHAN AND P. CHARTIER

where the Pk are the shifted Legendre polynomials,

Pk(x) = + (2)

orthonormal on the interval [0, 1],

[ Pk(x)Pe(x) dx = Ski, k ,t = 0 ,1 , . . . .Jo

It is well-known (see, for example, [3, 9]) that M (x) has real and distinct zeros and gives quadrature formulas of order greater than or equal to 2s — 2. The abscissas of the s-stage Runge-Kutta methods discussed in this paper are chosen to be these zeros for a given r and t < (s — l) /s , with the weights determined by the quadrature conditions B (s ) and known to be positive (see [3, 11]). Standard choices of the parameters (r, t) include the values (0, 0), (1, 0), (—1, 0) or (0, - 1) which lead respectively to the Gauss, Radau IA, Radau IIA, and various Lobatto III methods. The Runge-Kutta matrix A will be determined by the simplifying assumptions and other conditions associated with the desired properties of the method.

It transpires that the determination of the matrix A for a method with specified properties is most naturally and conveniently achieved through the s x s W transformation of Hairer and Wanner [8], defined by

W = [P0(c) Pi(c) Pa_ i(c )] ,

where c = [c i ,... ,cs]T. We consider the transformations X = W TB A W and J = W t B W , where B — diag(&i,. . . , 6S), and

S

X ki = bTPk^i{c)APi_i{c) = ^ 2 biPk-iicJdijPi-iicj),- k,£ = 1 , . . . , s,i,j=1

s

Jke = bTPk- 1(c)P£^1(c) = '^2biPk-i { c i)P i -1(ci), k j = l , . . . , s .1=1

The structure of these matrices is simplified considerably by the simplifying assumptions (see Butcher [3]). If / and g denote polynomials of degree less than q and r respectively, then

B(q) : kbTck~1 = 1, k = l , . . . , q <<=> bT/ (c ) = f f(x )d x .Jo

C(q) : kAck~ x = ck, k = l , . . . , q «=> A f(c) = f f(x )d x .Jo

D{q) : kbTC k~ 1A = bT - b TC k, k = l , . . . , q

bT f(c)A = bT f f(x ) dx.Jc

E(q,r) : £(k + i)bT ck~ l Acl~ l — 1, fc = l , . . . , g ; £ — \

bT f(c)Ag(c) = f(x ) g(y) dy j dx.

CLASSIFICATION OF HIGH-ORDER IMPLICIT RUNGE-KUTTA METHODS 153

The notation used here is f(d ) — [ f ( d i ) , . . . , f ( d s)]T and f ( D ) = d ia g (/(d i),. . . , f(d a)) for d = De and D = diag(di,. . . , da).

The quadrature conditions B (2s — 2) and orthonormality immediately yield

J = diag(l,. . . ,1,/x), (3)

where fi = 6T (Ps_ i(c ))2. Clearly n > 0, and (i = 1 if and only if B(2s — 1) holds.We consider methods that satisfy the simplifying assumptions B (2s — 2), C(s — 2),

D(s — 2) and E (s — 1, s — 1). These conditions imply order 2s — 2 and, together with the integrals,

if k — 0,r Pk(x)dx = p w + § * (# > .Jo [£k+1Pk+i(y) - £ fcPfc- i(y ) , if A: = 1,2,

(4)

where l /£ k = 2\/4k2 — 1, yield

X =

r 12 - 6 0 0 0 0 '

6 0 0 0 0

0 £2 0 0 0 0

0 0 0 0 —£ s-2 00 0 0 •• 6 - 2 0

.0 0 0 0 6 - iM7 V

:)APS- i (c ) and 7 , 5 G R. Once X is specified, t

(5)

determined by the transformation

A = W X W t B , r—1 (6)

Our discussion will be based on the matrix X or X since it depends on the 5 parameters r, £,7 , S, u, where V — v /ii2, which can be used to characterize the various properties of interest and the order conditions.

In Section 2 we assemble various results for the convenience of later discussion. Necessary and sufficient conditions for a method of the class considered to have a Pade approximation for its stability function are given in Section 3. The properties of algebraic stability, symmetry, symplecticity and stiff accuracy are introduced in Section 4 and their inter-relationships discussed. In Section 5 we consider partitioned Runge-Kutta methods as a means of achieving both stiff accuracy and symplecticity for Hamiltonian problems.

2. Some Preliminary Results

In the proofs of the results that follow, the use of B (2s — 2) and orthonormality will be implicitly assumed.


Theorem 2.1. Let the abscissas c \ , . . . , cs be the distinct zeros of M (x ), and the weights 61, . . . ,bs be determined by B(s). Then the following relations hold forn = bT{p,- 1(c))2:

1 1 st 1. fl = 1 —S — 1

2. ul = 1 + P S_ I ) \ ( 2s - 1 )bTc2‘ ~2 - 1) .

3. - 1) - p ; / ) 2 (2sbTc2a~1 - 1).

P roof.1. From the definition of Ps-i (x ) and orthogonality we have

Eliminating Ps{x) from the 3-term recursion relation for the shifted Legendre polynomials,

s£sPs(x) - ( x - i J P s_i(x) + (5 - 1)£s_ iP s_ 2(x) = 0,2,

and from the equation M (x) = 0 then yields

x P s - i ( x ) = ( \ - -SV ■■■■■) P s—1 (x) + ( S - 1 - S t ) C s - l P s - 2 ( x ) .2 2(2s — 1)

Now substituting for cPs_i(c) in bTcs~ 1Ps-i(c ) gives

bTcs~ 1Ps- 1(c) = (s - 1 - st)Cs-ibTcs~2Ps- 2(c).

Expressing cs~2 as a linear combination of the Pfc(c) for A; = 0 ,. . . , s — 2, then yields

b T c S ~ 2 p ’ - M =

The first result for fi then follows on simplification.

2. Let xs_1 = Ylk=o dkPk(x), where l /d a- i = \/2s - 1 i?* !?)■ Then

ds- 1fi - bTcs~ 1Ps-i(c )

= bTcs~1 (Ps_i(c) - + d~\bTc2s~2

= f x s 1 (Ps_i(x ) - c^J^x5 *) dx + ds} 1b1 Jo

= ds- i + dj} 1 (bTc2s~22s - l J ,

and the second result for fi follows.


3. From (1), M (c) = 0 and orthogonality we have

bT Pa-i(c)Ps(c) =

By definition (2) for Ps(c), and orthogonality, we also have

bT Ps-i(c )P s(c)

= -J2s - i ( ( 2S_ ‘ c ’ - 'P A c ) - (s - 1) ( 2 _ ^ j b Tc‘ ~2Ps(c)j .

Noting that

Ps (c) - V2s~+1 cs + V 2 s T l s ^ l ) 0* 1 an< -Fs (c) — >/2s + 1 cs

are polynomials of degree 5 — 2 and s — 1 respectively, application of B (2s — 2) and orthogonality then yields

bTcs~ 1Ps(c)

= v 2j + t ( ( 2; ) ( & v » - ‘

Substituting these expressions into the expression for bTPs-i(c )P s(c) with the use of result 2 then gives result 3.

□Remarks 2.2. Result 1 shows that /i > 0 since t < (s — 1 )/s , and that fi — 1 if and only if t — 0. Thus t = 0 and /i = 1 for Gauss and Radau methods, while t — — 1 and /I = (2s — l ) / (s — 1) for Lobatto methods. Result 2 then verifies the equivalence of B(2s — 1), fi = 1 and t — 0, while result 3 verifies the equivalence of B{2s) and r = t = 0. Hence, the theorem verifies that t = 0 and r — t = 0 are respectively necessary conditions for order 2s — 1 and 2s.

Theorem 2.3. For an s-stage Runge-Kutta method having distinct abscissas given by the zeros of M {x) with t < (s — l) /s , and a matrix X given by (5),

v = bTPs_i(c)AP s-i(c ) = i fi{6 - 7 ) + (2s - 1) .

Proof. Let x s~ x = Xlfc=o dkPk{x), where dk — x s~ 1Pk(x) dx. Noting that the matrix X is tridiagonal and skew-symmetric except for the elements X n = 1/2, X ss = v, X SjS- i = £s_ i/i7 and X a_ i jS = - £ s_i//<5, we have

bT cs 1 Acs 1 = d2_ i X ss + d^Xn + ds- i d s- 2( x s- i js + X SiS_ i)

= <% -iV + ^dl + d s - x d s ^ s - i ^ i l ~ S).

12s — s

2s - 1 s — 1 bTc2s~2

2s - 1


Now since d0 = 1 /s , l /d s_ i = y/2s - l ( 2°_?), and ds- 2 = da_ i /(2 fs_ i), the result for v follows. □

Rem ark 2.4. u — (6 — ry)/(2fi) if and only if bTcs~ 1Acs~ 1 = 1/(2s2).Theorem 2.5. For an s-stage Runge-Kutta method having abscissas given by the distinct zeros of M {x) with t < (s — l ) /s , and a matrix X given by (5),1. C{s - 1) < = ^ 7= 1.

2. D ( s - l ) «<=> 6= 1.

o \ , x 1 + * ~rfi3. C(s) <=> 7= 1, d = ------- , v—

4. D (s ) •<=>• <5= 1, 7=

fj, ’ 2 (2 3 -1 ) '

1 + i r/i/z ’ 2(2s - l ) ‘

P roof.1. Since C(s — 2) is assumed,

C ( s - l ) <*=> A P s_ 2 ( c) = [ Ps- 2{x)dx.Jo

Thus if C(s — 1) holds, then X S)S_i = £s-i/u and 7 = 1. Conversely, if 7 = 1 then X S;S_i = Cs-iV and = £s_ i. Now W TB P s_ 2(c) = es_ i, andC(s — 1) holds, sinceAPa_ 2(c) = W X W TBPs- 2(c) = W X e s- 1

£s—l^s £,s—2&s—2i if S > 2£s_ies + \es- i , if s = 2

= W

£ s - l P S- l ( c ) - £s- 2Ps- 3{c), if s > 2

t s - i P s -i ( c ) + | P .-2(c), if S = 2= / Ps- 2{x)dx.

Jo

2. Since D(s — 2) is assumed,

D(s — 1) «= » b1 PS- 2{C )A = bT [ Ps- 2{x)dx.Jc

If D (s — 1) holds, then X s_ i js = — £s_i/i4 and S = 1. Conversely, if <5 = 1 then x s_i,, = —£s- i , and D (s — 1) holds, since

bTPs- 2{C)A - bT Ps- 2(C )W X W t B = eTs_ 1X W T B

( ( 6 - 2 2 - 6 -1 e * )W TB , if s > 2

I (—i e? - i — ^ -1 eJ )W TB, if s = 2

'Z s-2P s -3(C) - 6 - iP s- i(C ), if s > 2

- ± P s - 2(C ) -t ;s - i P s - i ( C ) , a 8 = 2= b

bT f Ps- 2{x)dx. Jc


3. Now upon substituting for Ps(c) from M (c) = 0, we have

C(s) <=> C (s — 1) and APs-i(c ) = f Pa-i (x ) dx = £sPfl(c) - £s_ iP s_ 2(c)Jo

= -6 -l(l+ t)P .-2 (c ) - ^ P,-l(c).

If C(s) holds, then C(s — 1) implies 7 = 1 by result 1. Moreover, =—£s_ i ( l + 1) and X ss = —r/i/(4s —2). Hence the results. Conversely, 7 = 1 implies C (s — 1) by result 1. Also, X s_ i>s = — £s- i ( l + t)/fi and X ss = —r/((4 s — 2)/i) by hypothesis. Then C (s ) holds, for

APs_i(c) = W X W TB P s- 1{c) = W X /j,ea

= - W/ ( 6 - 1(l + *)es- i + 2(2sr_ 1) es)

= —£ s-i(l + t)P s-2(c) — 2 25 — l)

4. On substituting for PS(C ) from M (C ) = 0, we have

D{s) <=> £> (s- 1) and bTPs- 1{C)A

= bT f Ps-i (x )d xJc

= bT { ( U -i P -2{ C ) - Z aPa{C))

= (z .- i ( l + t)P.-2(c) + J _ P.-i(c)) B.

If .D(s) holds, then D(s — 1) implies 6 = 1 by result 2. Also, =6 - i ( l + t), X as = rfi/{As — 2) and the results follow. Conversely, <5 = 1 implies D (s — 1) by result 2. By hypothesis, = £s- i (1 + *)//-* andX ss = r /((4 s — 2)/i). Now D (s ) follows for, as in the above proof of D(s - 1),

bTPs- 1{C )A = bTPs- 1 (C )W X W TB = [ieTs X W TB

= ( 6 - 1(1 + t)eJL1 + 5^ L _ ^ ) ^ fl

= ( 6 - i ( l + t)P.-a(c) + 2(2/ _ x)P ,-i(c )) B.

□Remarks 2.6. Each of the conditions C(s — 1) and D (s — 1) reduces the number of parameters by one, while either C (s ) or D(s) reduces the number by three. If both C (s ) and D (s ) hold, then by result 1 of Theorem 2.1 we have r — t = u — 0, and fj, = 7 = 5 = 1, yielding the Gauss method of order 2s. If C(s) and D(s — 1) hold, then t = 0, /z = 7 = <5 = 1, v — — r/(4s — 2) and the order is at least 2s — 1. If C (s — 1) and D (s ) hold, then we have the same conclusions except that v = r/(4s — 2).


3. Pade Approxim ations

Runge-Kutta methods with stability functions given by the diagonal and the first two sub-diagonals of the Pade table for exp(z) are known to be ^4-stable as are algebraically stable methods. It is therefore of interest to establish conditions for a method, especially one that is not algebraically stable, to have one of these particular stability functions. In this section we determine such conditions for a method defined by (1) and (5).

The (A:,ra)-Pade approximation to exp(z) is given by Nkm(z)/Dkm(z), where

and satisfies Nkm{z)/Dkm{z) = exp(z) + 0 (zk+m+1) as 2 —> 0.Using the fact that the Gauss method has an X-matrix, X G, given by (5) with

H — 'y — S = 1 and u = 0, and that its stability function is the (s, s)-Pade approximation, we find that

N h k = i,fc—l + z 2£ k - i N k - 2 , k - 2 , k = 2, . . . , s ,

Nss = det (I + z X G),

D k k = D k ~ i , k - i + z 2^ l _ 1D k - 2, k - 2, k = 2,. . . ,s,

D ss = det (7 - z X G).In general, the stability function of a Runge-Kutta method can be expressed as

_ det (7 - z(J ~ lX - d e l ) )[Z) det(7 - z J - 'X )

Theorem 3.1. The following are necessary and sufficient conditions for an s - stage Runge-Kutta method having abscissas given by the distinct zeros of M {x) with t < (s — l ) /s , and satisfying B(2s — 2), C(s — 2), D(s — 2) and E (s — 1, s — 1) to have a stability function equal to one of the Pade approximations to exp(,z).

(s, s)-Pade : v — 0, fi'yS = 1.

(s — 1, s — I)-Pade : u = 0, 76 = 0.

(s - 1, s) -Pade : - = 1 fvyS = 1./x 2(2s — 1), _ N v 1 _ 2s - 1(s — 2, s)-Pade : - = —------—, fjt'y6 = ------ - .

fi — 1) s — 1

P roof. By hypothesis, X is given by (5) and we have

det(7 — z(J _1X — e\e{)) = ^1 - j-zJ iVs_ i )S_i + z2( 1s_ 1^ 8N a- 2,s -2, (7)

det(7 - zJ ~*X ) = ^1 - ^ z j -I- z2^2_ 1h^6D s^2,s -2- (8)


Equating these expressions to the corresponding ones for the (s,s), (s — 1, s — 1),

The conditions for degree s — 1 in the numerator and degree s in the denominator are given by

These are also necessary and sufficient conditions for R(oo) = 0 and for an A-stable method to be L-stable. The conditions for degree s — 1 in both the numerator and denominator, and for degree s —2 in the numerator and degree s in the denominator, are those for the (s — 1, s — 1) and the (s — 2, s)-Pade approximation respectively.

4. Properties and their Inter-Relationships

The equivalent conditions for algebraic stability [1, 2, 6, 18] are given by Hairer and Wanner [8],

where P is a permutation matrix such that Pc = e — c and Pb — b. We remark that the symmetry condition c* = 1 — cs+\-i holds if and only if r = 0. Now c* and1 — Ci are zeros of M (x ) if and only if c* is a zero of M (x ) and M ( 1 — x) if and only if Ci is a zero of (—1 )sM (x ) — M ( 1 — x). Applying the symmetry property, P fc (l-x ) = ( -1 )kPk{x), the latter expression equals 2 ( - l ) srPs_i(x) and the result follows.For stiff accuracy [12] it is easy to show that r = — 1 — t must hold and that the equivalent conditions are given by

(s — 1, s) and (s — 2, s)-Pade approximations then gives the stated conditions. □

The coefficients of zs and zs_1 in (7) are given by

while those in (8) are given by

( » - ! ) ( * - 2) )

V

B A + A TB - b b T > 0 X + X T — e ie j > 0.

The equivalent conditions for symplecticity [5, 13] follow directly,

B A + A TB - bbT = 0 X + X T - = 0. (10)

(9)

The equivalent conditions for symmetry [14, 17] are given in Chan [4],

A + P A P T = ebT ^ X u = i , X M = 0 if k + £ & 2) is even, (11)


Theorem 4.1. Suppose an s-stage Runge-Kutta method has abscissas given by the distinct zeros of M {x ) with r = — 1 — t and t < (s — 1 ) /s , and satisfies B(2s — 2), C(s — 2), D(s — 2) and E (s — 1, s — 1). Then the method is stiffly accurate if and only if 7 = 1 and V = 5/(4s — 2).

Proof. The method is stiffly accurate if and only if e ^ W X = e j . By hypothesis, cs = 1 is a zero of M (x) so that

S

eTs W = [ 1 v/3 ••• x/27^3 y /2 s = l\ = ^ \/2k - 1 e j,fc=l

and, moreover, X is given by (5). With J — d iag(l,. . . , 1,/i), and X =we have X = X except for X SjS_i = £s- i 7 , and X ss — V — v /y ? .Expressing the matrix I as s column vectors,

X = [fie2 + \e\ 6 e3 ~ £ ie i ••• C s-iies - £s_ 2es_2 Ves - £s_i<5es_i] ,

and noting that e J W X has a first component equal to 1, a A;-th component for k = 2 ,. . . , s — 2 given by & V 2k + 1 — £k-i V2k — 3 which vanishes, and has the last two components given by

£s_ i7\/2s — 1 — £s_ 2\ /2s — 5 and V\/2 s — 1 — £s_i<5\/2s — 3,

we then have e J W X = e j if and only if 7 = 1 and V = <5/(4s — 2). □

The class of s-stage methods with matrix X given by (5) will be denoted by C and defined by the quintuple of parameters,

C = | (r, t, 7 , 6, u) € R5 : t < S~~~~ jSubclasses can thus be defined in accordance with specific properties. By (9) the subclass of algebraically stable methods, denoted by Caig.stab., is defined by

^alg.stab. — {(^"j G C . 7 ^ ^ 0 } .

By (10) the subclass of symplectic methods, denoted by CSympiec.> is defined by

^symplec. — {(^*> t , 7> v ) G C . 7 &■> V 0 } •

By (11) the subclass of symmetric methods, denoted by CSymmet., is defined by

^symmet. — 1 1 0 £ C . T 0, V 0} .

By Theorem 4.1 the subclass of stiffly accurate methods, denoted by Cstiff.acc.) is defined by

'stiff.acc. = \ (fit, 5, v) C . r 1 £, 7 ^ 2(2s - 1)

These definitions lead directly to the following relations among the various subclasses of C.

CLASSIFICATION OF HIGH-ORDER IMPLICIT RUNGE-KUTTA METHODS

Proposition 4.2.1. Csymplec. C Calg.stab.-

2. CSym piec.^CSymmet. ~ C aig.stab.^C Symmet. = {(0, t, <7, (7, 0) : t < (s l) /s , (7 £

3. C-symplec. f'l C sym m et. = {(^j 0) • ^ ^ 0? 1 )/Sj <7 £ M }.

4- Csymplec. C stiff.acc. =

5- Cstiff.acc. Csym m et. = {(0) 1? 1 )0 ,0 )} .

6- C stiff.acc. H Calg.stab. = {(“ 1 ~ t , t , 1, 1, l/(4 s - 2)) : t < [s - l)/s} .

7. cstiff.acc. n Calg.stab. = { ( “ 1 ~ t , t , 1, <5, <5/(4« — 2)) : t < (s - l) /s , S ± 1}.

8- Csym plec. H Calg.stab. flC Sym m et. =

9- C stiff.acc. f'l C aig.s tab. f'l C Sym m et. ~

F ig u re 1. Various subclasses of C


Remarks 4.3.1. By result 2 the s-stage symplectie, symmetric and algebraically stable methods

is a two-parameter family. Putting t = 0 gives (0,0, cr, cr, 0), a one-parameter family based on Gauss-Legendre quadrature satisfying B(2s). The special case a = l gives the Gauss method of order 2s when both C (s ) and D (s ) hold. Putting t = — 1 gives a one-parameter family based on Lobatto quadrature. These are the Lobatto IIIS methods [4], (0, — l,cr, a, 0), of order 2s — 2 which include Lobatto HID in the special case a = 1 when C{s — 1) and D (s — 1) hold. The methods with a = ± \ /(s — l)/(2 s — 1) and <7 = 0 have stability functions given by the (s, s) and the (s — 1, s — 1)-Pade approximations respectively. The Lobatto HID method has a stability function not given by a Pade approximation. Other methods based on quadrature formulas given by values of t < (s — l) /s , t ^ 0 ,-1 , also include those satisfying C(s — 1) and D(s — 1) in the special case cr = 1.

2. By result 3 the s-stage symplectic and algebraically stable but nonsymmet- ric methods is a three-parameter family. Putting t = 0 and r = ±1 gives (±1,0, cr, cr, 0), a one-parameter family based on Radau I (r = 1) and RadauII (r = —1) quadrature and satisfying B(2s — 1). The special case cr = 1 gives the Radau IB and IIB methods [16] of order 2s — 1 and satisfying C(s — 1) and D(s — 1). Both these have the (s, s)-Pade approximation as stability function. The other methods based on Radau I and II quadrature, (±1,0, —1, —1,0) and ( ± 1, 0, 0, 0, 0), have stability functions given respectively by the (s, s) and the (s — 1, s — 1)-Pade approximations. Other choices of t < (s — l ) /s with t ^ 0 can yield methods satisfying C(s — 1) and D (s — 1) for the special case a = 1 but are only of order 2s — 2.

3. By result 5 the s-stage Lobatto IIIA, (0, —1,1,0,0), is the unique stiffly accurate and symmetric method. It is not algebraically stable but is ^4-stable with stability function given by the (s — 1, s — 1)-Pade approximation.

4. By result 6 the s-stage stiffly accurate and algebraically stable methods are a one-parameter family satisfying C(s — 1) and D (s — 1). If t = 0, C {s ) holds and gives the Radau IIA method of order 2s — 1 defined by ( — 1,0,1,1, l/(4 s — 2)), while t = — 1 gives the Lobatto IIIC method of order 2s — 2 defined by (0, — 1,1,1, l/(4 s — 2)). These have stability functions given respectively by the (s — l,s ) and the (s — 2,s)-Pade approximation. Other members of this family have stability functions given by the (s — 2, s)-Pade approximation.

5. By result 7 the stiffly accurate but non-algebraically stable methods include those based on Radau II quadrature, ( — 1,0,1, S, J/(4s — 2)), and on LobattoIII quadrature, (0, —1, l,S ,S /(4s — 2)). In both cases the choice 5 = 0 leads to the (s — l ,s — 1)-Pade approximation. The choice S = (2s — l) /(s — 1) in the Radau II case leads to the (s - 2, s)-Pade approximation, while the choice 5 = (s — l)/(2 s — 1) in the Lobatto III case leads to the (s — 1, s)-Pade approximation. These choices all give ^4-stable methods.


5. Symplectic Partitioned Runge-Kutta Methods

Stiff accuracy is a desirable property of Runge-Kutta methods in many applications to differential equations involving algebraic constraints. Unfortunately, result 4 of Proposition 4.2 shows that its condition is incompatible with the symplectic condition. Although nonsymplectic, stiffly accurate methods are nevertheless of interest in the construction of symplectic partitioned Runge-Kutta methods in applications to Hamiltonian problems.

Let (A, b, c) be a Runge-Kutta method and (A, b, c) its symplectic dual of a partitioned Runge-Kutta method satisfying the symplectic condition,

B A + A t B - bbT = 0. (13)

For nonzero weights the condition is equivalent to

A = ebT — B ~ 1A t B = (eeT - B ~ 1A T)B. (14)

Theorem 5.1. (Sun [15]) Suppose that a Runge-Kutta method with coefficients (Hj, bi ^ 0, and distinct C{, satisfies the simplifying assumptions B(p), C(q) and D (r). Then the partitioned Runge-Kutta method (A , b, c )-(A , b, c) is symplectic and satisfies

Sqh(to) = 0 (h ”+1), Sph(to) = 0 (h ”+ l),

with an order rj = min(p, 2q + 2, 2r + 2, q + r + 1).

For constrained Hamiltonian systems, Jay [10] has shown that the following additional simplifying assumptions have an important role in determining the order conditions.

C C (Q ): AAck~ * = _ , k = 2, . . . , Q .

......,

We determine necessary and sufficient conditions for a partitioned Runge-Kutta method to satisfy C C (s) and D D (s).

Theorem 5.2. If the method (A ,b,c) satisfies B(p) and C(q) (respectively D (r )) with p > max(g,r) then its symplectic dual (A,b,c) satisfies D(q) (respectively C(r), provided the weights are nonzero).

Proof. If (A ,b ,c ) satisfies C(q), then premultiplying (14) by bTC k~ x for each k = 1, . . . , q yields


Hence (A, b, c) satisfies D(q). If (A, b, c) satisfies D (r ) then postmultiplying (13) by C fe-1e for each k = 1, . . . , r gives

= 6(6TC fc-1e) - A TB C k~ 1e

= [

= ( ^ Tc ,fc) = ^ BC,fce-

Hence (A, 6, c) satisfies C'(r) since B is nonsingular. □

Lemma 5.3. Let (A, b, c) 6e an s-stage Runge-Kutta method with positive weights satisfying C( l ) and B(Q), and let (A ,b,c) be its symplectic dual. Then for2 < Q < s, C C {Q ) if and only if D D (Q ).

P roof. By (14), C( l ) and B (Q ) we have for each k = 2, . . . , Q

AAck~2 = A(ebT - B ~ x A TB)ck~2 = — - A B ~ lA TBck~2,k — 1

bTC k~2A A = bTC k~2A(ebT - B ~ 1A TB ) = %- - bTC k~2A B ~ X A TB ,k

bTCk~2AA = bl - ^ £ - + (BAAck~‘2)T- k k — 1

Hence the result follows easily from the last relation. □

Rem ark 5.4. If C(Q) and D(Q — 1) hold, then so do C(Q — 1) and D(Q) by Theorem 5.2, and it follows that both C C (Q ) and D D {Q) hold.

Theorem 5.5. Suppose an s-stage, s > 2, Runge-Kutta method (A ,b ,c ) has abscissas given by the distinct zeros of M (x ) with t < (s—l) /s , and satisfies B (2s—2), C(s — 1), D(s — 2), and E (s — 1 ,s — 1). If (A ,b,c) is its symplectic dual, then r = —2(2s — l)n5v and t + 1 = fi52 are necessary and sufficient conditions for CC (s) o rD D (s).


= w

Proof. From (14) and (6) we have for s > 3,

AAP s_ 2(c)

= - A B ~ 1A TBPs_ 2{c)

= - W X W TB W X TW TBPs_ 2(c)

= - W X J X Tes- 1

= W~XJ(£s- i 6es — Cs-2es- 2)= W X ( £ s- i n 6 e s ~ C s - 2 e s - 2 )

Cs— Cs — l^ s —l) Cs—2(Cs—2&s — 1 Cs—3 -s—3)) if S > 3 Cs-in6(vea - Cs-i£es_ i) - Cs-2(Cs-2es- i + §es_2), if s = 3

_ fCs-lM ^-Ps-l(c) ~ (C ?_ l^ 2 + C2- 2)^ s-2(c) + Cs-2Cs-3^s-4(c), if S > 3

\ca- i / i ^ p s- i (c ) - (C2- 1 2 + C?-2)-P*-2(c) - ^Cs-2Ps-3(c), if s = 3

Similarly, for 5 = 2 we find that

A AP s_ 2(c) = C s - i + P s - i ( c ) - ( C s - i M 2 - ^ P s - 2 (c )-

Now by C(s — 2) (or D (s — 2)), C(s — 1) and the use of definition (2) for Ps_2(c),

A A P ,.2(c) = V 2 ^ 3 ( 2; ; 24) ( A A c -> - ^ Y ) ) + [ ( jP f t - a ( * )< f c )

By (4) the integral evaluates to

C s -lC s P s (c ) - (C ? - l + C ?-2 )- P s - 2 (c ) + C s -2C s - 3 P s - 4 (c ) , if 8 > 3,

C s -lC s P s (c ) - (C ?_ l + es- 2)P s-2(c) - |C s - 2 P s - 3 (c ) , if * = 3,

^ C s -lC s P s (c ) + |C s - l P s - l ( c ) + { \ ~ C ? - l ) P a - 2 { c ) , if S = 2,

and it thus follows that in all cases AAcs~2 = c8/ (s(s — 1)) if and only if the c* are zeros of

£sPa(x) - fiSuPa-i (x ) + Cs-i(M 2 - l)P s_ 2(ar).

Comparison with (1) then yields the stated conditions for CC(s) and, by Lemma 5.3, D D (s). □

Rem ark 5.6. The Lobatto IIIA-IIIB pair (see Jay [10]) is an appropriate method for constrained Hamiltonian problems.

Acknowledgem ents. This work was carried out during the visit of R.P.K. Chan to IRIS A. He wishes to acknowledge the kind hospitality of the Institute and B. Philippe in particular. Both authors wish to thank Ernst Hairer for his kind suggestions.


References

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17. G. Wanner, Runge-Kutta methods with expansions in even powers of h, Computing, 11 (1973), 81-85.

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R.P.K. ChanDivision of Science and TechnologyTamaki CampusThe University of AucklandPrivate Bag 92019AucklandN E W ZEALAN D

chan® mat h. auckland. ac. nz

P. Chartier IRISACampus de Beaulieu Rennes 35000 FRAN [email protected]

mailto:[email protected]

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