Higher-order operator splitting methods for

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Higher-order operator splitting methods for deterministic parabolic equations A. T. Sornborger

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Department of Mathematics and Faculty of Engineering, University of Georgia, Athens, Georgia, 30602 Published online: 16 Jul 2007. To cite this article: A. T. Sornborger (2007): Higher-order operator splitting methods for deterministic parabolic equations, International Journal of Computer Mathematics, 84:6, 887-893 To link to this article: http://dx.doi.org/10.1080/00207160701458294

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International Journal of Computer Mathematics Vol. 84, No. 6, June 2007, 887–893

Higher-order operator splitting methods for deterministic parabolic equations Downloaded by [University of California Davis] at 13:57 29 May 2013

A. T. SORNBORGER* Department of Mathematics and Faculty of Engineering, University of Georgia, Athens, Georgia 30602 (Received 30 August 2006; revised version received 17 January 2007; accepted 30 March 2007 ) The Sheng-Suzuki theorem states that all exponential operator splitting methods of order greater than 2 must contain negative time integration. There have been claims in the literature that higher-order splitting methods for deterministic parabolic equations are unstable due to this fact. We show stability for a class of higher-order splitting methods for integrating deterministic parabolic equations. We note that problems with backwards time integration will still exist for stochastic integration methods for which information is lost and backward timesteps become ill-defined. Therefore, completely positive splitting methods, such as those developed by Chin, still have an important place. We present numerical results from first-, second-, third- and fourth-order methods showing that the error becomes increasingly small as the order increases. Keywords: Operator splitting; Parabolic equations; Sheng-Suzuki theorem; High order methods; Nonreversible systems AMS Subject Classifications: 65N12; 35K45; 35K50; 65Y20

1.

Introduction

Operator splitting methods [1] are used in many circumstances for integrating ordinary and partial differential equations. Particularly in the symplectic case (under the rubrik ‘symplectic methods’), they have enjoyed wide use due to their retention of the phase space characteristics of Hamiltonian systems. Advances have also been made recently in the application of these methods to equations with Lyapunov integrals [2]. In a 1989 paper [3], Sheng showed that higher-order (order greater than two) operator-splitting methods must contain operators which integrate backwards in time. This result has subsequently been found by others [4–6] and is now known as the Sheng–Suzuki theorem. Negative time integration does not cause difficulties for Hamiltonian systems. However, for stochastic equations, information is irrecoverably lost at each timestep and backwards integration becomes impossible. Due to the Sheng–Suzuki theorem, effort has been spent to avoid backward time integration in higher-order methods by explicitly including exponentials of commutators [7] in splitting methods. In the intermediate *Email: [email protected]

International Journal of Computer Mathematics ISSN 0020-7160 print/ISSN 1029-0265 online © 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207160701458294

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case of deterministic parabolic equations, where diffusive effects destroy time-reversal symmetry yet there is a deterministic (non-stochastic) equation describing the evolution, there have been claims that the backward integration steps of higher-order methods cause instability [3, 7]. In this paper, we present a stability analysis for a class of exponential operator splitting methods and a counter-example demonstrating that, although it is true that higher-order splitting methods all must have negative time evolution operators, nevertheless, they are not all unstable for the integration of linear parabolic equations. We present simulation results for third- and fourth-order operator splittings which are stable and demonstrate increased accuracy relative to lower-order splittings. First, we give a brief recapitulation of how our integration methods were derived [6]. Then, we show that they are stable. Finally, we integrate a discretized version of the linear equation ∂φ ∂(xφ) ∂ 2 φ = + 2 ∂t ∂x ∂x

(1)

in one dimension. We can rewrite this equation as φ˙ = Dφ

(2)

where D is the discretization to fourth order in #x of the operator ∂x x + ∂x2 . Integrating for a time #t, we have the solution φ(t + #t) = e#tD φ(t).

(3)

Here, D = D1 + D2 where D1 is the operator ∂x x discretized to fourth-order in #x and D2 is the operator ∂x2 also discretized to fourth-order. We integrate equation (1) in time using a fourth-order Runge–Kutta scheme and take the results of this unsplit scheme as the solution to compare the split schemes against. The particular equation used here does not require a splitting, exact solutions are known. We have used it because the two operators do not commute, and the continuous problem has an exact solution to check our methods against. Splitting methods are particularly valuable for equations where an analytical solution is not available for a sum of operators, but analytical (preferably quickly calculable) solutions are known for each operator in the sum. We note that the splitting methods used here can be used for arbitrary sums of operators, i.e. D1 + D2 + · · · + DN , where the Di are all operators, none of which need commute with the others (see [6]). In particular, these methods may be used for linear parabolic equations, where exactly solvable diffusive terms as well as other easily integrable terms can be split.

2.

Operator splitting methods

Our !methods were derived in the following manner [6]. We want to express the operator # "N exp n=1 An as a product of individual factors exp(An )’s. There will be many possible !" # N combinations that approximate exp n=1 An to a given order. We choose to investigate

Higher-order operator splitting methods for deterministic parabolic equations

889

combinations of fundamental units of the form %α $ aA1 aA2 e e . . . eaAN

(4)


with parameters a and α allowing for transposes of the entire product as well as raising all the exponentials in the fundamental unit to the same power. Choosing to search for approximations combining fundamental units of this form has the benefit that all approximations will be valid for an arbitrary number of non-commuting terms N . By iterating the Campbell–Baker–Hausdorff formula, we find an expression for the fundamental unit (4) in terms of a single exponential ∞ & %α $ aA1 aA2 p αa p BN e e . . . eaAN = exp

(5)

p=1

p

that defines the BN in terms of the An . Here, p is an exponent on a and a label on the matrices p BN . We take α = ±1. Combining i = 1, . . . , I fundamental units with parameters ai and αi and using the Campbell–Baker–Hausdorff formula, we find     + , ∞ ∞ & & & p p p p X X α1 a1 BN  · · · exp  αI aI BN  = exp σ I BN . exp  p=1

p=1

(6)

X

p

Here, the BNX are commutators of the BN s. X represents a label pq · · · rs, where pq···rs

BN pq···rs

BN

. .. - p - q ≡ BN , BN , . . . , BNr , BNs · · · .

(7)

is of order p + q + · · · + r + s. Up to 5th order, we can take X ∈ {1; 2; 3, 12; 4, 13, 112; 5, 14, 23, 113, 221, 1112}.

(8)

The σIX are defined in terms of αi and ai by (6). After some calculation, we obtain the following equations for the σIX . p

σI =

I &

p

αi ai

(9)

i=1

for p = 1, . . . , 5, I

. 1 p q 1 & q−p - p 2 pq p σI = − σI σI + ai (σi ) − (σi−1 )2 2 2 i=1

(10)

for pq = 12, 13, 14, 23, ppq

σI

I

. 1 p pq 1 p 1 & q−p - p 3 q p = − σI σI − (σI )2 σI + (σi ) − (σi−1 )3 ai 2 6 6 i=1

(11)

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A. T. Sornborger

for ppq = 112, 113, 221, and I


. 1 1 1 1 & - 1 4 1 σI1112 = − σI1 σI112 − (σI1 )2 σI12 − (σI1 )3 σI2 + ai (σi ) − (σi−1 )4 . 2 3 24 24 i=1

(12)

" X 1 For approximations to exp( N n=1 An ), all σI must be zero, except for σI , which should be greater than zero. There are no non-trivial solutions to equation (9) with p = 3 when the product αi ai is positive for all i. Therefore, for third-order methods, αi ai must be negative for at least one i. This implies that there must be at least one operator with negative time evolution in operator splitting methods of order 3. Similarly, with p = 3 and p = 4 in equation (9), it can be proved that fourth-order methods must have at least two operators with negative time evolution [6]. The operator splitting methods given below were obtained by#solving (9–12) for approxi!" N mations with integral coefficients to the operator exp n=1 An up to first-, second-, thirdand fourth-order. We use the notation (#t) to represent $ #tD1 #tD2 % e e

and (#t)T to represent

$ #tD2 #tD1 % e . e

(13)

(14)

So, for example, our first-order method

$

is represented by

e#tD1 e#tD2

%

(#t)

(15)

(16)

and our second-order method $ #tD1 #tD2 % $ #tD2 #tD1 % e e e e

is represented by

(17)

(#t)(#t)T .

(18)

(#t)T (#t)(#t)(#t)(#t)T (−2#t)T (#t)(#t)(#t)

(19)

Our third-order method is

where the reader will note the operator with negative time evolution, and our fourth-order method is (#t)T (#t)(#t)T (−2#t)(#t)T (#t)T (#t)T (#t)T (#t)(#t)T (#t)(#t)(#t)(#t)(−2#t)T (#t)(#t)T (#t). This fourth-order method has two negative time evolution operators.

(20)


3.

891

Stability


THEOREM 3.1 All methods of the form (6) are stable for the integration of deterministic parabolic equations if the fundamental unit is stable. Proof We prove the theorem for matrix operators. We start by assuming that the fundamental unit, a first-order operator splitting method, is stable. This can be checked with standard techniques. A stable first-order method will be non-increasing /$ %α // / aA1 aA2 (21) / e e · · · eaAN / = eaα& ≤ 1,

" "d k k where & ≡ N n=1 k=1 λn ≤ 0, d is the dimension of the matrices {An }, λn is the kth eigenvalue of the nth matrix and we have used the fact that |M T | = |M| for a matrix, M. To determine stability for the full method, we check that it is also non-increasing / I / /0 / "I I / a i A 1 ai A 2 ai A N αi / ···e ) / = e i=1 ai αi & = eσ1 & ≤ 1, (22) / (e e / / i=1

but this will always be true since σ1I > 0 for our methods.

!

For the third-order method used here, σ1I = 6, and for the fourth-order method, σ1I = 12. From the above theorem, we see that, although some integrations in the operator splitting method are backwards in time, the overall integration has an amplification factor that is less than one. Possible instabilities arising from the negative time integration are cancelled and the higher-order methods are stable. Below, we demonstrate this fact numerically and show that our methods are also more accurate as the order increases.

4.

Numerical results

We integrated the split operators (16), (18), (19), (20) using fourth-order spatial finite differences and a fourth-order Runge–Kutta algorithm [8]. As stated above, we compared the evolution as integrated by the split (spatially discretized) operators to the unsplit (spatially discretized) operator. 2 As initial conditions, we took φ(x, t0 ) = −e−t0 xe−1/2x with t0 = 0. We discretized space with #x = 0.05, #t = #x 2 /60 with 500 gridpoints. These parameters gave us a stable first-order split operator integration. At the boundary we took derivatives to be constant. We checked our space and time discretization by verifying that the discrete solution integrated using the unsplit operator was a good approximation to the solution of the con2 tinuous equation (i.e. φ(x, t) = −e−t xe−1/2x ). The solution was integrated for 100, 000 time steps. In figure 1(a), we plot the logarithm of the spatially integrated error 1 (23) E = |φunsplit (x) − φsplit (x)| dx for each method versus the logarithm of time (counted in units of the number of timesteps). The first-order spatially integrated log-error is uppermost, with the second-order log-error beneath it, and so on down to the fourth-order log-error at the bottom. Note that the fourthorder error never gets much larger than the numerical roundoff accuracy (≈ 10−15 ) of the


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A. T. Sornborger

Figure 1. Panel (a), The logarithm of the spatially integrated error is plotted as a function of the logarithm of time. The error for the first-order operator splitting method is topmost, followed successively by the second-, third- and fourth-order errors. The fourth-order error is of order the numerical roundoff accuracy of the computer. Panel (b), The unsplit, exact solution is plotted as a function of the logarithm of time from 0 to 100, 000 timesteps. Panels (c)–(f), The spatial distribution of the logarithm of the absolute value of the error for first- through fourth-order operator splitting methods as a function of the logarithm of time.

computer. In figure 1(b), we plot the exact, unsplit solution in space as a function of log-time. In figure 1(c)–(f), we plot the spatial distribution of the log-error as a function of timestep for each of the methods from first-order (c) to fourth-order (f). The fourth-order log-error looks less structured than the first- through third-order log-errors because it is at the edge of the numerical roundoff accuracy of the computer. By the end of the simulation, all errors are of order the numerical roundoff accuracy since the solution relaxes to zero along the entire grid (as seen in figure 1(b)). For the class of methods discussed here, we expect the nth-order error per timestep from the splitting scheme E n ∝ R n+1 (#t)n+1 (see Appendix 3 in [6]), where R n+1 is a coefficient proportional to an (n + 1)st-order commutator. For our equation, R 1 ∝ [D1 , D2 ] = −2∂x2 . Remember that for our simulation #t = #x/60, giving a first-order error E 1 ∝ 10−5 , and the errors of higher-order methods lie proportionally beneath the first-order error. In the figure, we see that the errors are as expected in that they lie proportionally beneath each other (in log units) and their spatial support is where |∂x | is relatively large (see figure 1(b)). 5.

Conclusions

We find that the class of operator splitting methods given by the factoring (6) are stable for deterministic parabolic equations provided the fundamental unit (4) is stable. There is


893

no instability induced by negative time-evolution in the splitting. We then demonstrate that (1) the errors are spatially distributed in the way that would be expected given the form of the commutator of the terms in the deterministic parabolic equation that we chose to integrate as an example, and (2) this class of methods is as accurate as we would expect it to be, given the values of the error calculated for a given splitting, for deterministic parabolic equations.


References [1] Strang, G., 1968, On the construction and comparison of difference schemes. SIAM Journal of Numerical Analysis, 5, 506–517. [2] McLachlan, R., Quispel, G. and Robidoux, N., 1998, Unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions or first integrals. Physical Revue Letters, 81, 2399. [3] Sheng, Q., 1989, Solving linear partial differential equations by exponential splitting. IMA Journal of Numerical Analysis, 9, 199–212. [4] Suzuki, M., 1991, General theory of fractal path integrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics, 32, 400–407. [5] Goldman, D. and Kaper, T., 1996, Nth-order operator splitting schemes and nonreversible systems. SIAM Journal of Numerical Analysis, 33, 349–367. [6] Sornborger, A. and Stewart, E., 1999, Higher-order methods for simulations on quantum computers. Physical Revue A, 60, 1956–1965. [7] Chin, C.A., 2004, Quantum statistical calculations and symplectic corrector algorithms. Physical Revue E, 69, 046118. [8] Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P., 1994, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd edn (Cambridge: Cambridge University Press).