Parallel general linear methods for stiff ordinary differential and differential algebraic equations

Transcription

1 ~-~H-~-~ ~ APPLIED ~ NUMERICAL MATHEMATICS ELSEVIER Applied Numerical Mathematics 17 (1995) Parallel general linear methods for stiff ordinary differential and differential algebraic equations J.C. Butcher ~"*, P. Chartier b,l Department of Mathematics The Unicersity of Auckland, Auckland, New Zealand b I.R.I.S.A., Rennes, France Abstract For numerical methods of Runge-Kutta type, the value of thc stability function at infinity, R(~), is a key factor to the solution of stiff ODEs and DAEs. In particular, methods with R(~) = 0 are especially attractive. The aim of this paper is to make some preliminary comparisons of two recently introduced approaches to the construction of L-stable parallel general linear methods, that differ essentially by the behaviour they require at infinity. 1. Introduction Since parallelism across the method makes sense only in a multi-stage environment, it is natural to consider Runge-Kutta methods. However, Iserles and Ncrsett [4] have shown that the order of a Runge-Kutta method cannot exceed s + 1 where s is the number of sequential stages. In an attempt to circumvent this barrier, van der Houwen and Sommeijer [6] have proposed a parallel iterative scheme for the nonlinear system to be solved at each step. Another possibility is to consider the wider class of "general linear methods" (GLMs). Two recent approaches to parallel GLMs are the "Generalized Backward Differentiation Formulas" [3] and DIMSIM methods of type 4 [1]. If we denote by yen-~l the r-block vector of output values at step number n - 1 (these are also the input values to step number n) and by Y the s-block vector of stage values, then one step of these method can be written as Yi[n-l] = h ~aijf(yj[n-1])jr_ ~ u i l y j,[n 1] i = l,...,s, (1) j - I ) 1 y}n]: h ~ bijf(yj[n I])_~_ ~ "'ijyj"[n- l], i = I,..., r, (2) j - 1 j = l where the initial value problem under consideration is y'(x)=f(y(x)). b) * Corresponding author. Telephone: Fax: butcher(c mat.auckland.ac.nz. l Telephone: (33) Fax: (33) chartier(a'irisa.fr /95/$ Elsevier Science B.V. All righls reserved SSDI (95 )

2 214 J.C. Butcher, P. Chartier /Applied Numerical Mathematics 17 (1995) The r output values have here a natural interpretation in the sense that y!'-11, i = 1,..., r, is assumed to be an approximation at point x, _ 1 of order p to the quantity p ;i(x) = E k = 0 In Section 2, we will present methods that are an obvious extension to type-4 methods (see [1]) and that furthermore satisfy B = Vil. Section 3 will consider "pure" type-4 methods, characterized by A = AI. In particular, we shall construct L-stable examples. Since for both methods the evaluation of the stages are independent, they are natural candidates for parallel implementation. In Section 4, preliminary numerical results will be given. Finally, further possible developments will be sketched. (3) 2. Pure A-V methods We consider here methods for which A is a nonconstant diagonal matrix, A = diag(a 1..., As). In this case, the amplification matrix, obtained by a standard linear stability analysis, is of the form M ( z ) = V + z B ( I - z A ) - I u. While the condition p ( M ( ~ ) ) = 0 is sufficient to achieve a good damping of the errors, it may seem even more desirable to require M(~) = 0. This ensures that, in the situation of differential algebraic equations, the numerical solution lies on the constraint manifold. If we arbitrarily choose U = I, this is equivalent to B = VA whenever A is non-singular, and the order conditions take a particularly simple form: Theorem 1. Let us consider a generalized type-4 method of the form [Avil / ] (4) and let p = r - 1 = s - 1. Denote by l:(x) the jth Lagrange polynomial based on the abscissae c:, j = 1...., s; the method is of order p and stage-order p iff V = L - AL', where the (i, j) elements of L and 12 are respectively given by lj(1 + c i ) and Ii(1 + c i ). (6) (5) Proof. Since the order and the stage-order do not differ, the conditions for order p = r - 1 can be written as V e cz = ( I - z A ) e <]l+c,z --t- O ( z r ), (7) where c = (c c,) T, 1[ = ( ) T, e <z = (exp(clz),..., exp(csz)) T and e (~ +<)z = (exp((1 + el)z)..., exp((1 + cs)z))) v. This implies that VP(c) = P ( 1 + c ) - A P ' ( ~ + c ) (8) for all polynomials of degree less than r - 1. Taking P ( x ) = lj(x) for j = 1..., s then gives the coefficients of V. []

3 J.C. Butcher, P. Chartier /Applied Numerical Mathematics 17 (1995) It can be noted that the usual B D F methods (Backward Differentiation Formulas) form a special subset of the set of methods described by the previous theorem. They can be obtained by taking c i = - ( s - 1) + i, i = 1,..., s, and A = d i a g ( 0,..., 0, As), where A, is chosen so as to achieve stage-order p = r and order p. However, it is a well-known fact that BDFs are not A-stable for orders greater than 2. Other choices of the ci's seem to offer only tiny improvements on the usual BDFs (see [2]). As an illustration of this fact, we give below a m e t h o d based on the abscissae c = [ - 3, ~, 1] v, with stage-order 3 and order 3, r = s = 3: l! ( {) (9) In a parallel environment and for a large problem, this m e t h o d has a computational cost similar to a B D F m e t h o d but better properties (its error constant is C = 0.22 and it is A(86.7 )-stable 1 compared to respectively - ~ and 86 for the B D F of order 3). However, it can definitely not be considered as a major competitor of BDF. This seems to suggest that A-stable m ethods of order p --- r = s and stage-order p might not exist and that good methods within that class are difficult to construct. On the other hand, it has been shown [3], that other choices of the A/'s may lead to E-stable methods up to very high orders (at least 12). The optimal choice of the Ai's and c/'s in the general situation still remains an open question. As an example, we take c = [0, 1 34 ' 1ft. A numerical search for A-stable methods with small coefficients has led to the m e t h o d l~/ _ ) (18(/ ( l (10) Numerical evaluation of the coefficients in the V matrix (B = VA) gives V = (11)

4 216 J.C. Butcher, P. Chartier /Applied Numerical Mathematics 17 (1995) If the vector of input values y["-q lies on the exact solution 37(x,_ 1) at the point xn_l, then the error after one step of method (10) is of the form with Now let h4(a y(4)(xn_l) ) + O(hS), (12) A = [ I T ~ ~ ~ J " [ T U ~ [ ~ ~ , ~ J be the normalized left eigenvector of V associated with the eigenvalue 1. The local truncation error 6 of method (10) is given by "~h4~(4)(,," (~ = h 4 ( L : T A ) y ( 4 ' ( X n - 1 ) + O(kS) = ( 1 ~ 2 1 ' " -Y ~,~n-l) + O(h5) (13) A natural way to get an estimation of this local error is to consider a linear combination of the scaled derivatives of the stage values 4 (14) E Oihf(Y,[~-lJ). i = l It is then easily seen that 4 E Oihf(Yii In-l]) = ( 1 ]'" ~ 2 ] h 4 w ( 4 ) ( y* y, ~ n - 1) + O(hS), ( 1 5 ) i = 1 if and only if E,4:10i = O, E4: loici = O, E4_10,c2 = 0 ' (16) Et loic3 = " " Solving the linear system (16) leads to I i],[i:l = ( ~ 0 (17) Thus, an asymptotically correct error estimator is 15216;90706 {hf( Y1 ["- 1]) _ 2hf(YLzn-1]) + 2hf(y~n-,]) _ hf(y4 ~"- 1])}. (18) It can be computed without any additional evaluation of f.

5 J.C. Butcher, P. Chattier/Applied Numerical Mathematics 17 (1995) Pure type-4 methods By a "pure type-4 method", we will mean a method in which r = s and the matrix A is of the form AI. For reasons of simplicity and for good stability at 0, we will assume that the matrix V is of the form V = ]u x, where vt]i = 1. This assumption will guarantee that o-(v)= {1, 0}, as for many methods such as those of Adams type, but will also go a little further. In fact the Jordan canonical form for V is diag(1, 0,..., 0). It is convenient to arbitrarily choose U as the unit matrix. This assumption means that the r quantities computed in a step and on which the stages of the following step depend are precisely equal to these later stages minus h times the corresponding stage derivatives. A standard linear stability analysis of a DIMSIM leads to the stability matrix M = V + z B ( I - z A ) ~U. With the special choice we have made for A and U, this simplifies to M ( z ) = V B, 1 - A z Z with characteristic polynomial of the form d e t ( I w - M ( z ) ) = P ( w, z) = (1 -Az)--~fi(w, z), where the polynomial /3(w, z) is given by / 5 ( w, z ) = ~ C k ( z ) ( 1 - A z ) ~ ~w ~-k, k = () and the polynomials C k have degrees k for k = 1, 2...., s, with Co(z) = 1. On the assumption that V has rank 1, many coefficients in these polynomials vanish. In fact, C~(z) = O(z~-l), for k = 1, 2,..., s. The remaining coefficients in C 1, C2....,C s are functions of the elements in the vector v and can be adjusted to fine tune the coefficients in the stability polynomial. One natural choice amongst the many available is to require that C k has degree only k - 1, for k = 1, 2,..., s. This will imply that the spectral radius of M is zero when z = ~. Thus C k = - a k z k ', where ak, k = 1, 2,..., s, as well as A are completely determined by the order conditions. Having determined the stability polynomial in this way, there is still some freedom in the 1 choice of the vector of abscissae c. If for r = s = 3 we choose c = [0, 3, 1] T, the method takes the form [) 0 1 (I 0 A o (~ 1 o ~ ( JA + 276A 2 ) [) A ~( A 2 ) J,(83 208A + I(12A 2) ~,(79 174A +72A 2) ~( A - 72A 2) 6~( A +72A 2) " ] ~ ( _ A + I I O 4 A 2 ) 1( 199~497A 240A 2) /4( A_816A2) 1(79 174A~72A2) 1( A 72A 2) 61( A A 2 ) ] [ ~(ii0 280A+138A -~) ~a( A 241)A 2 ) J6( A-+2()4A 2 ) 1(79 174A+72A2) 1( A 72A 2) 6~( A A 2 ) J where A is a zero of the polynomial A - 18A 2 + 3A 3. An analysis of the stability in this (19)

6 2 ] ~ J.C. Butcher, P. Chartier/Applied Numerical Mathematics 17 (1995) shows that this method is A-stable (and hence L-stable) if A is chosen as the zero approximately equal to The local truncation error 6 of this scheme can be estimated as described in the previous section (see Eqs. (12) and (13)) = 2~4( A - 168A2)h4y )( x,,_ l) + O(hS). (20) Getting an error estimator is however slightly more complicated, since only three stages are at our disposal. Hence, we seek a linear combination of the scaled derivatives hf(y}~-ll), i = 1, 2, 3, of the stage values and of the incoming values y!~-l], i = 1, 2, 3, such that 3 3 y ~ - - [n O, Yi 1] [n - + ~_,Oihf(Y~ q ) = ~ ( A A 2 ) h a y ( 4 ) ( X n _ l ) + O ( h 5 ). i=1 i=1 (21) Assuming that the local truncation error vector has been chosen so as to lie along the preconsistency vector, a Taylor expansion of the left-hand side gives the following system of equations El} l~)i = O, E;'= O, Jr- E~ lljktei= J i,j = O, 2~_ loici -~- 2 i = I(~)zE) = lbi,j(cj -- 1) = O, z~= ~o~: + z ~ _, ~,, z ~ =, b,,~(~ - 1) 2 = o, (22) zl,ei~;~ + z~=,,~,zi~= ~b/j(~- 1) ~ = 3!. 2 4 ( A - 168A2). In order to get a reliable estimation for small values of h, we furthermore impose that 3 0 = Y~j= 1 j 0. A few calculations eventually lead to the following values of the ~bi's and 0 i s, A A 2 3(1-2 A ) _, _168 &2 = 3(1-2A) ' 4'3-1 ¼(-1-4 A ) ] 2A ¼(1-4 A ) (23) so that an asymptotically correct error estimator reads as A A 2 ) 3 ( 1-2A) {-Yln 1]+2y~,, q_y~n l]+¼(_l_4a)hf(y~n 1]) + 2Ahf(Y[2 n 1]) + ¼(1-4A)hf(Y[3n-a])}. (24)

7 J.C. Butcher, P. Chartier /Applied Numerical Mathematics 1 7 (1995) Unfortunately, this method is unattractive from another point of view. Numerical evaluation of the coefficients in the B and V matrices gives their approximate values as ] B = /, (25) ] ] V = /. (26) ] The large values of many of the coefficients in these matrices suggest that this method is unsatisfactory because of the large roundoff errors that might result from its use. It is possible that this might be remedied by other choices of the c vector. We now consider an alternative method based on the choice of the c vector c = [ - 1, 0, 1] T. With the same value of A, the new m e t h o d has the following tableau 0 0,~ A " ~ ( A + 156A 2) {( A 6(Ia 2) 1~(65-166A + 84A2) ] 16(57-149A + 78A 2) ~( A 60A 2) ~ ( A A 2) [ t ( a a 2) ½( a - 60a z) ~ ( ,~ a z) Numerical values for this case are as follows {( A A 2) ½( A 18A 2) I~( 16 39A + 18A2) ~3( A 18h 2) ~ ( A + 18A 2 ) ½ ( A A 2 ) 0 ~6(34 57A + 18A 2) 1 t~,(34 57A + 18A 2) ~(34-57A + 18A 2) (27) B = V = ] , ] To estimate errors for this method, one can use the following formula, obtained by solving the system of equations (22) where the right-hand side of the last equation has been replaced by the appropriate value, (28) (29) A - 168A 2 ) { _ y ~ 11+2Y~2 n 11_y~n-11+3( _ 1-2 A ) f i f ( y l P ' ) + 2ahf(Y2 t"-'l) + ½(1-2a)hf(Y~"-l])}. (30) 4. N u m e r i c a l t e s t s Methods (10) and (27) were tested on the following problems:

8 220 J.C. Butcher, P. Chartier /Applied Numerical Mathematics 17 (1995) Prothero-Robinson Problem (Stiff ODE). y ' ( x ) = ( y ( x ) - s i n ( x ) ) / E + c o s ( x ), Y0-- 0, x ~ [0, 11], (31) with exact solution y(x) = sin(x). We chose here e = Modified Kaps Problem (Index-2 DAE). y ' i ( x ) = - ( 2 + e - 1 ) Y l ( X ) + e - l y ~ ( x ), y l ( O ) = l y'z(x) = - e 1-z'~'x), Y2(0) = 1, 0 =Y2(X) 2 - y, ( x ) e ~(1-c s~y~-y2(~)2)), z(o) = 1 with exact solution y, ( x ) = e -2~ y 2 ( x ) = e -~, z ( x ) = ~ + x In our experiments, e was set to 10-2 and a to Nonlinear Index-3 Problem (see [6]). y',(x) = 2yl(x)yz(x)z,(X)Zz(X), y,(o) = 1, y'2(x) = - Y l ( X ) Y 2 ( X ) Z 2 ( X ), Y2(O) = 1, Z'I(X)= ( Y l ( X ) Y 2 ( X ) +Z1(X)Z2(X))U(X), zi(o)= 1, Z'z(X ) = -Yl(X)y2(x)z~(x)u2(x), z2(0 ) = 1, O = y, ( x ) y 2 ( x ) - 1, u(o) = 1, x ~ [0, 1], (32) (33) x ~ [0, 11, (34) with exact solution y, ( x ) = e 2x, Y 2 ( x ) = e ~, z ~ ( x ) = e 2~, z 2 ( x ) = e -x, u ( x ) = e x (35) For comparison purposes, we have also included in our tests the R u n g e - K u t t a method of order 3 based on the Radau IIA abscissae. The systems have been integrated with fixed h. Numerical results for these problems are shown graphically in Fig. 1. The observed orders of convergence for the R u n g e - K u t t a method (2 for the y- and z-components and 1 for the u-component) are in agreement with those predicted by [5, T h e o r e m 6.1, Chapter IV]. 5. Conclusion Both methods solve the three test problems without difficulty and apparently without the order reduction found for the Radau IIA method. Our hope is to understand how the quality of the method depends on the spacing of the ci's and the spectral properties of M(0) = V and M(oo) = V - B A - 1 U. Our preliminary results, in which (10) performs especially well, suggests that M(oo) -- 0 is a desirable property. Method (10) also seems to benefit by having s > p, which results into an accuracy comparable to the Radau IIA method. This has the additional

9 2. C. Butcher, P. Chartier /Applied Numerical Mathematics 17 (1995) Prothero-Robineon Problem -~/ i '! i '!,. y ~ 2[ Modified Kaps Problem : y-cemlaenerit i o - > =,. F ~. m, ~ a..... ~ :..... ~ -4 =, ~ a d a ~ l ~ _>,, +-" ~ = ;..'-* [ ~"'- 5. r I... ~ "..... i '"-,,.,t,-,'., /2 -i -&.0',-&.6 o 02,2.,.3, -3' -2.,.,'5 Ioglo(h) leglo(h) Modified Kaps Problem : z-component Nonlinear index 3 problem : y-component 2 i -1!, ~ > F~sl m o t t ~ d -3 o => F+~t method ~ - "=> RaeautlA3 o :, r " ; ~ : : / :\ / : o / w 4" ~ " 2 ", ~ :..,K i ~ -3.5, -3 i -25 i -2 i -15 4, i i -2 I -1.5 I leg1d{h) ~ l o l h ) Nominear index 3 problem : z-componem Nonlinear index 3 problem : u-component -1 2 [ I -3" O => FirSl method ~ : => Second fnelilod ~. ~ / /, / -3 >= N ~ ~"..... ~' i / / 4,, i i I -3 ~2.5-2 iogl0(h), Fig. 1., J i i L Iogl0(h)

10 222 Z C. Butcher, P. Chartier /Applied Numerical Mathematics 17 (1995) advantage to allow an easy and reliable error estimate. On the other hand, method (27) shows a more regular behaviour and this is likely to result from having V of rank one. Further work on these questions is proceeding. References [1] J.C. Butcher, Diagonally-implicit multi-stage integration methods, Appl. Numer. Math. 11 (1993) [2] J.C. Butcher and P. Chartier, The construction of DIMSIMs for stiff ODEs and DAEs, Report Series, University of Auckland, Auckland, New Zealand (1994). [3] P. Chartier, L-stable parallel One-block methods for ordinary differential equations, SlAM J. Numer. Anal 31 (1994) [4] A. Iserles and S.P. Ncrsett, On the theory of parallel Runge-Kutta methods, IMA J. Numer. Anal. 10 (1990) [5] L.O. Jay, M6thodes du type Runge-Kutta pour des 6quations diff6rentielles alg6briques d'index 3 avec des applications aux syst~mes hamiltoniens, Thesis, University of Geneva, Geneva, Switzerland (1994). [6] P.J. van der Houwen and B.P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control, J. Comput. Appl. Math. 29 (1990)