S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal., 36:475-491, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....di erential equations. In particular, Hamiltonian ordinary di erential equations can be solved by so called symplectic or variational integrators [9, 17, 28] In addition to exactly preserving the symplectic form, these integrators have been shown to conserve energy over exponentially long times [1, 7, 8, 26]. Such conservation properties are of crucial importance in celestial mechanics, molecular dynamics, and other application areas. As a continuous system and its discretization will generally diverge arbitrarily far from each other as time advances, results on approximate energy conservation for ....

S. Reich. Backward error analysis for numerical integrators. SIAM Journal on Numerical Analysis, 36(5):1549-1570, 1999. 38

....map, a subject of much recent interest. For such maps, it is possible to show the existence of a perturbed Hamiltonian function whose exact dynamics replicate the numerical solution, up to an error term exponentially small in the timestep (O(e 1=Dt ) for a long (but finite) interval in time [1 3]. The presence of this approximate conserved quantity appears to confer greater numerical stability on the numerical simulation. In practice, when integrating a system with smooth dynamics using a symplectic integrator with sufficiently small stepsize, one observes that the energy of the numerical ....

.... new time variable, say t, satisfying = g(q; p) 1) A major disadvantage of the general Poincare transformation is that it mixes the variables so that an explicit symplectic treatment of the extended Hamiltonian is no longer possible, and we are compelled to use implicit symplectic methods (see [3,7]) A first order method can however be made semi explicit in the case of an N body system with separated Hamiltonian function T (p) V (q) but a second order method is often required for accuracy reasons. A more serious defect associated with the Poincare transformation is that it allows the ....

S. REICH, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., (submitted 1996).

....will be addressed in the sequel. 1.2 Methods based on composition and splitting When a numerical method is used to approximate the solution of (1) at t = h, it is common to represent it by a map h;F called the h flow of the method. We write y 1 = h;F (y 0 ) By using backward error analysis [5, 19], one can in some important cases, at least formally, associate the h flow of a method with the flow of a perturbed vector field F h such that h;F = exp(h F h ) A popular way [23, 20] of constructing high order numerical integration methods is to compose h flows h;F = h 1 ;F ffi ....

S. Reich. Backward error analysis for numerical integrators. Technical Report SC 96-21, Konrad-Zuse Zentrum fur Informationstechnik Berlin, 1996.

....If the discretization of the reparameterized equations preserves a geometric structure, then we can view the resulting method as a structure preserving variable stepsize method. For example, when the Poincar e transformation is used, then a symplectic stepsize variation method can be obtained [29, 14]; e.g. a symplectic, second order method is obtained by solving (3.3) 3.4) using the implicit generalized leapfrog discretization (Lobatto IIIA B Partitioned Runge Kutta method) 14] If a Sundman time transformation is positive and invariant under the involution p Gammap, then the rescaled ....

S. Reich, Backward error analysis for numerical integrators, preprint, 1996.

....h . A method is symplectic if Psi h is symplectic. Following are some reasons for using symplectic integrators for long time dynamics: i) Backward forward error analysis: the numerical solution is nearly the exact solution of a nearby Hamiltonian system on a time interval of length O(1=h) See [2,9,10,15,18,23]. Although 1=h is theoretically a long time, in practice, it may be orders of magnitude shorter than the integration interval. Nonetheless, there is no evidence that the lifespan of the backward error analysis cannot be extended to much longer times. ii) Energy conservation as an error ....

S. Reich, Backward error analysis for numerical integrators, Technical Report SC 96-21, Konrad-Zuse-Zentrum fur Informationstechnik Berlin, July 1996.

....differential equations, the modified equations are also Hamiltonian if and only if the integrator is symplectic. The existence of a modified, or shadow [4] Hamiltonian is an indicator of the validity of statistical estimates calculated from long time integration of chaotic Hamiltonian systems [18]. In addition, the modified Hamiltonian is a more sensitive indicator than is the original Hamiltonian of drift in the energy (caused by instability) Evidence for the existence of a Hamiltonian for a particular calculation can be obtained by calculating modified Hamiltonians and monitoring how ....

....T . The modified equation for an integrator Phi h applied to this system is Hamiltonian, i.e. f h = JH h;x (x) for some modified Hamiltonian H h (x) if and only if the integrator is symplectic [23, 20] The integrator is symplectic if Phi h;x (x)J T Phi h;x (x) j J . There is theoretical [2, 8, 18] and empirical evidence that x n = x h (nh) very small error for a very long time where x h is the solution for a suitably truncated Hamiltonian H h . In what follows we assume that H h is such a Hamiltonian and we neglect the very small error. If we plot total energy as a function of time for ....

S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal. To appear.

....concept known as backward error analysis in which one considers a modi ed di erential equation and provides error statements with respect to this modi ed equation. This has proved to be a very useful concept of error, especially in the the case of Hamiltonian systems (see e.g. 1] 15] 16] and [23]) This paper is organized as follows. In section 2 we present the shadowing setup that we will employ throughout this paper and in section 3 we present an improved xed point result that has two fewer norm bounds than the previous result. In section 4 we consider a reformulation of the given ....

S. Reich, \Backward error analysis for numerical integrators," preprint (1998).

.... and schemes for ordinary di erential equations that conserve Lie point symmetries [25] Furthermore, for some speci c problems such as Hamiltonian ODEs or time reversible systems, some excellent geometry preserving integrators have been developed which can be analysed using backward error analysis [36]. However, a general theory, applicable to a wide class of problems is still lacking. In this review we will look in more detail at some of the items above and then see how well numerical methods do when trying to reproduce some of the qualitative structures. 2 Symmetry In this section we will ....

S. Reich, \Backward Error Analysis for Numerical Integrators", SIAM J. Numer. Anal., 36, (1999), pp. 1549-1570.

....Casimirs, the nearby differential equation has the form y 0 (t) J( y)r H( y) y(t 0 ) y 0 ; 3.16) where H : O y0 R is a Hamiltonian function. The numerical method also conserves the energy H . Since H is a first integral of the exact solution it is also preserved by (3. 16) [21]. d dt H( y) rH( y) T y 0 = rH( y) T J( y)r H( y) n H; H ofi fi fi Oy 0 ( y) 0: 8 From this we can conclude that H is a first integral of the Lie Poisson equations. Assume that H is a Casimir. Then the function restricted to the coadjoint orbit is constant, and the ....

....methods presented in [26] the error in the Hamiltonian energy is bounded. Applying the backward error analysis results of [7] the Lie group methods are solving a nearby differential equation with solutions on the same coadjoint orbit. Moreover, the results by Hairer and Stoffer [11] and Reich in [21], combined with backward error analysis on Lie groups, imply that if the original problem is reversible, and the numerical method is reversible, then the perturbed problem is also reversible. KAM theory for reversible systems applied to periodic problems implies that, if the original differential ....

S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), pp. 1549--1570.

....the observed differences in stability of the methods. As an outcome of this analysis, we show in Sect. 3 how the oscillations can be attenuated by correcting the starting stepsize. Another extension of the Verlet scheme to variable stepsizes has recently been proposed independently in [4] and in [9]. In addition to being symmetric, it is a symplectic method, but it has the disadvantage of being implicit in g n 1 . A comparaison of the different extensions is given in [2] 2 Backward Error Analysis and Asymptotic Expansions For the choice g n = G(q n ; p n ) the method is a symmetric ....

S. Reich, Backward Error Analysis for Numerical Integrators, preprint, 1997.

.... of these properties are local behaviour near steady states, stable periodic orbits, homoclinic orbits [FS96] and energy conservation for symplectic integrators [BG94] San92] There is a vast literature on di erent aspects of backward error analysis for ordinary di erential equations, see [HL97] [Rei99] and the review article [HL99] and references therein. We will consider semilinear parabolic partial di erential equations, see e.g. Hen83] As a model take a system of reaction di usion equations with U = u 1 ; un ) D = diag(d 1 ; dn ) with d i 0, and F = f 1 ; fn ....

S.Reich, Backward error analysis for numerical integrators, SIAM Num. Anal. 36 (1999), 1549-1570.

....near to the identity discrete maps. An important property of the modified equations is summarized in the following meta theorem: the modified equations inherit integrals, symmetries, reversing symmetries and symplectic structure from the discretization. Special cases of this result are proved in [41, 18, 14, 36]) The case of a reversing symmetry has been considered in [20] When the discretization scheme is symplectic, i.e. z Phi T Deltat J z Phi Deltat = J; J = 0 I GammaI 0 ; then the meta theorem implies the existence of a perturbed Hamiltonian expansion H (k) Deltat = H ....

....methods. However, it is important to recognize that the bound in terms of O(exp( GammaK= Deltat) is only valid in an asymptotic sense, for small stepsizes (for which the error is very small) at large stepsizes, the energy variation is complex and unpredictable. For more details, see [3, 19, 36]. 5 Returning to the Figs.1d,e,f, we comment on the appearance of the graphs. For the symplectic Euler method it is easy to see that, H (1) q; Gammap) H (1) q; p) 2H(q; p) O( Deltat 2 ) 2.2) hence H (1) q; 0) H(q; 0) O( Deltat 2 ) and, using (2.2) together with ....

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S. Reich, Backward error analysis for numerical integrators, preprint, 1996.

....Some of these properties are local behaviour near steady states, stable periodic orbits and energy conservation for symplectic integrators [BG94] There is a vast literature on di erent aspects of backward 6 CHAPTER 1. INTRODUCTION error analysis for ordinary di erential equations, see [HL97] [Rei99] and the review article [HL99] and references therein. When considering special solutions we can also explain, why there has to be still some error kx n x(nh)k for any f for most numerical methods . One example is the discretization of homoclinic orbits, where a solution (t) converges to a ....

S.Reich, Backward error analysis for numerical integrators, to appear in SIAM Numerical Analysis (1999).

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S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal., 36:475-491, 1999.

....It has been shown that the preservation of the symplectic structure of phase space under a numerical integration scheme implies a number of very desirable properties. Namely, ffl symplectic methods preserve the total energy over very (exponentially) long periods of time up to small fluctuations [2, 11, 14] and ffl symplectic methods also conserve the adiabatic invariants of the problem under consideration [15] Note that the same results have not been shown for symmetric (time reversible) integration methods, although symmetric methods seem to perform quite well in practice. For a discussion of ....

S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal., submitted, 1997.

....are discretized in time by a generalization of the symplectic and time reversible Stormer Verlet method. The fact that the method is symplectic implies that the (finite dimensional) Hamiltonian (total energy) is very well preserved over long time intervals provided the step size is small enough [4, 20]. 2. The shallow water equations. The standard shallow water equations for a shallow homogeneous fluid in a coordinate system rotating at constant angular velocity f=2 are given by d dt h = Gammahr x Delta u; d dt u = Gammaf e z Theta u Gamma c 2 o rx h; d dt x = u: Here h is the ....

S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., to appear.

....1 U rpH (H Gamma fl)rp( 1 U ) dfl d = 0; dp d = Gamma 1 U rqH Gamma (H Gamma fl)rq ( 1 U ) dt d = 1 U : Here represents a fictive time. These Hamiltonian equations can be integrated using a symplectic discretization scheme. This idea has been explored recently by Reich [19] and Hairer [11] Although it is possible to obtain a first order explicit symplectic method for the extended phase space Hamiltonian, proposed higher order methods appear to require the work of several force evaluations per timestep, a serious drawback for large computations. Instead of treating ....

S. Reich, Backward error analysis for numerical integrators, preprint, 1996

....= 1 U rpH (H Gamma fl)rp( 1 U ) dfl d = 0; dp d = Gamma 1 U rqH Gamma (H Gamma fl)rq ( 1 U ) dt d = 1 U : Here represents a fictive time. These Hamiltonian equations can be integrated using a symplectic discretization scheme. This idea has been explored recently by Reich[22] and Hairer [12] Although it is possible to obtain a first order explicit symplectic method for the extended phase space Hamiltonian, proposed higher order methods appear to require the work of several force evaluations per timestep, a serious drawback for large computations. Instead of treating ....

S. Reich, Backward error analysis for numerical integrators, preprint, 1996.

.... j 1 Gamma j Deltat = dp j 1 dq j 1 Gamma dp j dq j Deltat = 0 ; 1.8) or dp j 1 dq j 1 = dp j dq j : 1. 9) This conservation property of a symplectic integrator gives rise to excellent long time behaviour which can be explained in terms of backward error analysis (see, for example, [3, 15]) For generalization to PDEs it is useful to abstract (1.1) 1.3) as follows. Let J be any skewsymmetric invertible matrix on R 2n . Then (R 2n ; is a symplectic manifold with (U; V ) hJU; V i where U; V 2 R 2n and h Delta; Deltai is the standard inner product on R 2n . A ....

S. Reich, Backward error analysis for numerical integrators, SIAM Numer. Anal. (1999) to appear.

....the convergence of long time averages along numerically computed trajectories. The anisotropic Kepler problem [15] will serve us as a numerical illustration. This problem requires the application of a symplectic variable step size method as first discussed by the author in the technical report [27] and independently by Hairer in [17] 2. The Modified Vector Field Recursion. Let us consider a smooth vector field d dt x = Z (x) 2.1) Z : U ae R n R n and its discretization by a one step method [19] xn 1 = Psi ffit (xn ) xn ffit (xn ; ffit) 2.2) We assume that Psi ffit : ....

....p(0) Then (5.9) 5.11) can be simplified to d d q = ae(q) M Gamma1 p ; d d p = Gammaae(q) r q V (q) d d t = ae (q) on the hypersurface of constant energy E = 0. This is a scaled vector field as desired but which is not Hamiltonian anymore. Therefore, as suggested by the author in [27] and independently by Hairer [17] the Hamiltonian equations (5.9) 5.11) are discretized by a symplectic method and e = H(q 0 ; p 0 ) For example, the equations can be discretized by the symplectic Euler method, i.e. q n 1 = q n ffi ae(q n ) M Gamma1 p n 1 ; p n 1 = p n Gamma ffi ae(q ....

S. Reich, Backward error analysis for numerical integrators, Preprint SC 96-21, KonradZuse -Zentrum Berlin, 1996.

.... approximation of the wave equation and subsequent symplectic integration in time will conserve this energy very well provided Deltat small enough [2] This also applies to the multi symplectic Gauss Legendre methods of x2 which follows from Proposition 2 and backward error analysis results in [2, 16]. Furthermore, there is also conservation of global momentum I(t) Z L x=0 I(z(x; t) dx = I(0) Z L x=0 I(z(x; 0) dx if z(0; t) z(L; t) periodic boundary conditions) Since I is a quadratic expression in v and w, Gauss Legendre collocation methods will preserve the discrete global ....

S. Reich, Backward error analysis for numerical integrators, SIAM Numer. Anal. (1999) to appear.

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S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal. 36 (1999) 1549--1570.

No context found.

S. Reich, Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, (submitted 1996).

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S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), pp. 1549-1570.

No context found.

S. Reich, Backward error analysis for numerical integrators, Technical Report SC 96--21, Konrad--Zuse Zentrum fur Informationstechnik Berlin, 1996.

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