MODIFIED DIFFERENTIAL EQUATIONS. Dedicated to Prof. Michel Crouzeix

Transcription

1 ESAIM: PROCEEDINGS, September 2007, Vol.21, Gabriel Caloz & Monique Dauge, Editors MODIFIED DIFFERENTIAL EQUATIONS Pilippe Cartier 1, Ernst Hairer 2 and Gilles Vilmart 1,2 Dedicated to Prof. Micel Crouzeix Abstract. Motivated by te teory of modified differential equations backward error analysis an approac for te construction of ig order numerical integrators tat preserve geometric properties of te exact flow is developed. Tis summarises a talk presented in onour of Micel Crouzeix. Résumé. Motivé par la téorie des équations modifiées analyse rétrograde de l erreur, une approce pour la construction de métodes numériques d ordre élevé préservant des propriétés géométriques du flot exact est développée. Ceci résume une présentation donnée en onneur de Micel Crouzeix. Introduction Modified differential equations in combination wit backward error analysis cf. te monograps [3], [5] form an important tool for studying te long-time beaviour of numerical integrators for ordinary differential equations. Te main idea of tis teory is sketced and, by inverting te roles of te exact and numerical flows, a new approac for te construction of ig order numerical integrators for ordinary differential equations is developed [1]. As an application, a computationally efficient and igly accurate modification of te Discrete Moser Veselov algoritm for te simulation of te free rigid body is presented [4]. 1. Modified equations for backward error analysis Consider an initial value problem ẏ fy, y0 y 0 1 wit sufficiently smoot vector field fy, and a numerical one-step integrator y n+1 Φ f, y n. Te idea of backward error analysis is to searc for a modified differential equation ż f z fz + f 2 z + 2 f 3 z +..., z0 y 0, 2 wic is a formal series in powers of te step size, suc tat te numerical solution {y n } is formally equal to te exact solution of 2, y n zn for n 0, 1, 2,..., 3 see te left picture of Figure 1. Tis work was partially supported by te Fonds National Suisse, project No INRIA Rennes, Campus de Beaulieu, Rennes-Cedex, France 2 Section de Matématiques, Université de Genève, 2 4 rue du Lièvre, CH-1211 Genève 4, Switzerland c EDP Sciences, SMAI 2007 Article publised by EDP Sciences and available at ttp:// or ttp://dx.doi.org/ /proc:072102

2 ESAIM: PROCEEDINGS 17 ẏ fy ż f z Backward error analysis numerical metod exact solution y 0, y 1, y 2, y 3,... z0, z, z2,... Modifying numerical metod ẏ fy ż f z exact solution numerical metod y0, y, y2,... z 0, z 1, z 2, z 3,... Figure 1. Backward error analysis opposed to modifying numerical integrators Te idea of backward error analysis was originally introduced by Wilkinson 1960 in te context of numerical linear algebra. For te integration of ordinary differential equations it was not used until one became interested in te long-time beaviour of numerical solutions. Witout considering it as a teory, Rut [9] uses te idea of backward error analysis to motivate symplectic integrators for Hamiltonian systems. In fact, applying a symplectic numerical metod to a Hamiltonian system ẏ J 1 Hy gives rise to a modified differential equation tat is Hamiltonian. Tis permits to transfer known properties of perturbed Hamiltonian systems e.g., conservation of energy, KAM teory for integrable systems to properties of symplectic numerical integrators. One became soon aware tat tis kind of reasoning is not restricted to Hamiltonian systems, and new insigt can be obtained wit te same tecniques also for reversible differential equations, for Poisson systems, for divergence-free problems, etc. A rigourous analysis as been developed in te nineties. We refer te interested reader to [3, Capter IX], were backward error analysis and its applications are explained in detail. 2. Modifying numerical integrators Backward error analysis is a purely teoretical tool tat gives muc insigt into te long-term integration wit geometric numerical metods. We sall sow tat by simply excanging te roles of te numerical metod and te exact solution cf. te two pictures in Figure 1, it can be turned into a means for constructing ig order integrators tat conserve geometric properties. Tey will be useful for integrations over long times. Let us be more precise. As before, we consider an initial value problem 1 and a numerical integrator. But now we searc for a modified differential equation, again of te form 2, suc tat te numerical solution {z n } of te metod applied wit step size to 2 yields formally te exact solution of te original differential equation 1, i.e., z n yn for n 0, 1, 2,..., 4 see te rigt picture of Figure 1. Notice tat tis modified equation is different from te one considered before. However, due to te close connection wit backward error analysis, all teoretical and practical results ave teir analogue in tis new context. Te modified differential equation is again an asymptotic series tat usually diverges, and its truncation inerits geometric properties of te exact flow if a suitable integrator is applied. Te coefficient functions f j z can be computed recursively by using a formula manipulation program like maple. Tis can be done by developing bot sides of zt+ Φ f,zt into a series in powers of, and by comparing teir coefficients. Once a few functions f j z are known, te following algoritm suggests itself. Algoritm 2.1 modifying integrator. Consider te truncation ż f [r] z fz + f 2z + + r 1 f r z 5 of te modified differential equation corresponding to Φ f, y. Ten, z n+1 Ψ f, z n : Φ f [r],z n

3 18 ESAIM: PROCEEDINGS defines a numerical metod of order r tat approximates te solution of 1. We call it modifying integrator, because te vector field fy of 1 is modified into f [r] before te basic integrator is applied. Tis is an alternative approac for constructing ig order numerical integrators for ordinary differential equations classical approaces are multistep, Runge Kutta, Taylor series, extrapolation, composition, and splitting metods. It is particularly interesting in te context of geometric integration because, as known from backward error analysis, te modified differential equation inerits te same structural properties as 1 if a suitable integrator is applied. A few known metods can be cast into te framework of modifying integrators altoug tey ave not been constructed in tis way. Te most important are te generating function metods as introduced by Feng [2]. Tese are ig order symplectic integrators obtained by applying a simple symplectic metod to a modified Hamiltonian system. Te corresponding Hamiltonian is te solution of a Hamilton Jacobi partial differential equation. Anoter special case is a modification of te discrete Moser Veselov algoritm for te Euler equations of te rigid body, proposed by McLaclan and Zanna [7]. Te general approac of Algoritm 2.1 is introduced and discussed in [1]. Example 2.2. For te numerical integration of 1 we consider te implicit midpoint rule y n+1 y n + f Te truncated modified vector field corresponding to tis metod is f [5] f f f f f f, f yn + y n f f f f f f f, f f f f f f, f f 1 2 f f f f, f + f f f, f f f f, f f, f 1 2 f3 f, f, f f f f 3 f, f, f f4 f, f, f, f and applying te midpoint rule to ż f [5] z yields a numerical approximation of order 6 for 1. At first glance tis modified equation looks extremely complicated and it is ard to imagine tat te modifying midpoint rule can compete wit oter metods of te same order. Tis is true in general, but tere are important differential equations for wic te evaluation of f [r] y is not muc more expensive tan tat of fy, so tat te modifying integrators of Algoritm 2.1 can become efficient. A first example is te equations of motion for te full dynamics of a rigid body see [1,4] and Section 3 below. As anoter example, consider te N-body problem, wic is Hamiltonian q p, ṗ Uq and as potential Uq U jk q j q k 1 j<k N te sum is over j and k, were U jk r is a scalar function U jk r 1/r for te gravitational potential and stands for te Euclidean norm. Here, q R 3N is composed by te position vectors q j R 3. Te vector field requires te computation of 7 Uq q j 1 k j N V jk q j q k q j q k, V jk r U jk r r 8 te sum being only over k. Te main observation is now tat every summand in tis expression depends only on two variables q j and q k. Terefore, many mixed iger derivatives vanis, and all expressions appearing in te modified equation 7 reduce to a sum over only one summation index. Typically, te computation of te square root in q j q k is te most time consuming part, and derivatives of U jk r can often be obtained

4 ESAIM: PROCEEDINGS 19 wit negligible cost wen computed togeter wit te value U jk r. In tis situation, modifying numerical integrators can be implemented efficiently. As noted by McLaclan [6], tis feature of N-body problems can be exploited also in an efficient implementation of implicit Runge Kutta metods. 3. Accurate rigid-body integrator based on te DMV algoritm As illustration of ow efficient modifying integrators can be, we consider te equations of motion for a rigid body, ẏ ŷ I 1 y, Q Q Î 1 y, were â 0 a 3 a 2 a 3 0 a 1 9 a 2 a 1 0 for a vector a a 1, a 2, a 3 T. Here, I diagi 1, I 2, I 3 is te matrix formed by te moments of inertia, y is te vector of te angular momenta, and Q is te ortogonal matrix tat describes te rotation relative to a fixed coordinate system. As numerical integrator we coose te Discrete Moser Veselov algoritm DMV [8], were te ortogonal matrix Ω n is computed from ŷ n+1 Ω n ŷ n Ω T n, Q n+1 Q n Ω T n, 10 Ω T n D D Ω n ŷ n. 11 Here, te diagonal matrix D diagd 1, d 2, d 3 is determined by d 1 + d 2 I 3, d 2 + d 3 I 1, and d 3 + d 1 I 2. Tis algoritm is an excellent geometric integrator and sares many geometric properties wit te exact flow. It is symplectic, it exactly preserves te Hamiltonian, te Casimir and te angular momentum Qy in te fixed frame, and it keeps te ortogonality of Q. Its only disadvantage is te low order two. Te tecnique of modifying integrators cannot be directly applied to increase te order of tis metod, because te algoritm 10 is not defined for general problems 1. It is, owever, defined for arbitrary I j, and terefore we look for modified moments of inertia Ĩj suc tat te DMV algoritm applied wit Ĩj yields te exact solution of 9. It is sown in [4] tat tis is possible wit s 3 y n + 4 s 5 y n d 3 y n + 4 d 5 y n Ĩ j I j Te expressions s k y and d k y can be computed by a formula manipulation package similar as te modified differential equation is obtained. Te first of tem are s 3 y n Hy n + I 1 + I 2 + I 3 Cy n, 3 I 1 I 2 I 3 6 I 1 I 2 I 3 d 3 y n I 1 + I 2 + I 3 1 Hy n Cy n, 6 I 1 I 2 I 3 3 I 1 I 2 I 3 were Cy 1 y1 2 + y2 2 + y3 2 and Hy 1 y y2 2 + y I 1 I 2 I 3 are te Casimir and te Hamiltonian of te system. Te pysical interpretation of tis result is te following: after perturbing suitably te form of te body, an application of te DMV algoritm yields te exact motion of te body. Truncating te series in 12 after te 2r 2 terms, yields a modifying DMV algoritm of order 2r. Example 3.1. We consider an asymmetric rigid body wit moments of inertia I 1 0.6, I 2 0.8, and I on te interval [0, 10]. Initial values are y0 1.8, 0.4, 0.9 T and Q0 is te identity matrix. Te implementation of te modifying DMV algoritm is done using quaternions as explained in [4]. Altoug Hy

5 20 ESAIM: PROCEEDINGS error angular momentum y order 2 order error rotation matrix Q order 2 order order 8 order order 10 cpu time order 10 order 8 order 6 cpu time Figure 2. Work-precision diagram for te DMV algoritm order 2 and for te modifying DMV integrators of orders 4, 6, 8, and 10. and Cy are constant along te numerical solution, we recompute te values of Ĩj in every step to simulate te presence of an external potential. We apply te DMV algoritm and its extensions to order 4, 6, 8, and 10 wit many different step sizes, and we plot in Figure 2 te global error at te endpoint as a function of te cpu times. Te execution times are te average of 1000 experiments. Te symbols indicate te values obtained wit te step sizes 0.1 and 0.01, respectively. Te pictures nicely illustrate te expected orders of te algoritms order p corresponds to a straigt line wit slope p. Muc more interesting is te fact tat ig accuracy is obtained more or less for free. Consider te results obtained wit step size 0.1. Te error for te DMV algoritm order 2 is more tan 20%. Wit very little extra work, te modification of order 10 gives an accuracy of more tan 11 digits wit te same step size. References [1] P. Cartier, E. Hairer, and G. Vilmart. Numerical integrators based on modified differential equations. Submitted for publication, [2] K. Feng. Difference scemes for Hamiltonian formalism and symplectic geometry. J. Comp. Mat., 4: , [3] E. Hairer, C. Lubic, and G. Wanner. Geometric Numerical Integration. Structure-Preserving Algoritms for Ordinary Differential Equations. Springer Series in Computational Matematics 31. Springer-Verlag, Berlin, second edition, [4] E. Hairer and G. Vilmart. Preprocessed Discrete Moser Veselov algoritm for te full dynamics of te free rigid body. Submitted for publication, [5] B. Leimkuler and S. Reic. Simulating Hamiltonian Dynamics. Cambridge Monograps on Applied and Computational Matematics 14. Cambridge University Press, Cambridge, [6] R. I. McLaclan. A new implementation of symplectic Runge Kutta metods. Submitted for publication, [7] R. I. McLaclan and A. Zanna. Te discrete Moser Veselov algoritm for te free rigid body, revisited. Found. Comput. Mat., 5:87 123, [8] J. Moser and A. P. Veselov. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Comm. Mat. Pys., 139: , [9] R. D. Rut. A canonical integration tecnique. IEEE Trans. Nuclear Science, NS-30: , 1983.