E. Hairer, Variable Time Step Integration with Symplectic Methods, Appl. Num. Math. 25, 1997, 219-227.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... 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 ....

E. HAIRER, Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), pp. 219--227.

.... in ae(q) However, in many cases the scaling function ae(x) can be greatly simplified and its evaluation is cheap compared to the evaluation of the force field F (q) Gammar q V (q) Independently of us, the same approach to symplectic variable step size integration has been formulated by Hairer [15]. Numerical example. As a numerical example, we look at the following modified Kepler problem: d dt q = p ; d dt p = Gammar q V (q) q; p 2 IR 2 , and V (q x ; q y ) Gamma 1 q (q x =10) 2 (q y ) 2 : The problem is non integrable and, in fact, the dynamics is chaotic, i.e. can be ....

Hairer, E., Variable time step integration with symplectic methods. manuscript, Geneva, 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 ....

....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 equations of motion will possess the time reversal symmetry, and use of an appropriate discretization scheme results in a time reversible variable stepsize strategy [37, 19, 18] We ....

E. Hairer, Variable timestep integration with symplectic methods, preprint, Dept. of Mathematics, Geneva, 1996.

....zero. Other papers have appeared discussing symplectic variable stepsize integration. Lee, Duncan, and Levison [7] provide an identical but less detailed discussion of this modification to leapfrog with emphasis on astrophysics applications, and they suggest different force decompositions. Hairer [4] describes a more general approach to symplectic variable stepsize integration by perturbing the Hamiltonian to one with an identical solution. However, his strategy does not use multiple time stepping and seems to be of limited use to molecular dynamics in which it is more natural to integrate ....

E. Hairer, Variable time step integration with symplectic methods, 1996, submitted.

....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 the extended ....

E. Hairer, Variable time step integration with symplectic methods, Appl. Num. Math., to appear

.... 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 the extended ....

E. Hairer, Variable time step integration with symplectic methods, preprint, 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 [36, 17]; 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) 17] If a Sundman time transformation is positive and invariant under the involution p Gammap, then the rescaled ....

....example, when the Poincar e transformation is used, then a symplectic stepsize variation method can be obtained [36, 17] 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) [17]. If a Sundman time transformation is positive and invariant under the involution p Gammap, then the rescaled equations of motion will possess the time reversal symmetry, and use of an appropriate discretization scheme results in a time reversible variable stepsize strategy [44, 25, 24] We ....

E. Hairer, Variable timestep integration with symplectic methods, Applied Numerical Mathematics 25, 219, 1997.

....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 : U ae R n R n is a smooth map ....

....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 n ) r q V (q n ) Gamma ffi ....

E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math., 25 (1997), pp. 219--227.

....= 0, p 2 (0) p (1 e) 1 Gamma e) corresponding to an orbit of period 2 and eccentricity e. This problem and its variable stepsize integration were carefully studied in [3] It was found that the fourth order Gauss Legendre method [7] combined with a symplectic variable stepsize strategy [6, 17] performs best among variable stepsize geometric integration methods of order two and four and that this method may also outperform standard integration software. In particular, it was also demonstrated that, despite its simplicity, the adaptive Stormer Verlet method was not competitive due to its ....

sc E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math., 25 (1997), pp. 219--227.

....a fictive time variable and fl is typically chosen such that e H is equal to zero along the desired solution. These differential equations can be integrated using a symplectic discretization scheme with fixed stepsizes in . This idea has been explored recently by Reich [25] and Hairer [15]. It was found by those authors that, in order to obtain a semi explicit symplectic method, a symplectic first order Euler method has to be used [13] The Poincar e transformation can also be applied to the constrained system (1.1) 1.3) and the resulting equations can be discretized by an ....

E. Hairer, Variable time step integration with symplectic methods, Appl. Num. Math., 25 (1997), pp. 219--227

....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) 2.2) 1 The general idea can already been found in the report [26] 2 Hairer Lubich [18] consider methods that can be represented by P series [19] Note that Runge Kutta and partitioned ....

....(5.26) 5.28) 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 energy shell 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 0 500 1000 1500 1 1.2 1.4 1.6 1.8 time 0 500 1000 1500 0.8 1 1.2 1.4 1.6 1.8 time 0 500 1000 1500 1 1.5 2 2.5 3 time 0 100 200 300 400 0.5 1 1.5 2 time (a) 2 1 0 1 2 5 0 5 X 1 0 500 1000 1500 10 10 10 8 10 6 10 4 10 2 10 0 time actual step size 0 500 1000 1500 4 2 ....

Hairer, E., Variable Time Step Integration with Symplectic Methods, Appl. Numer. Math. 25, 219--227, 1997.

.... variable, say , satisfying dt d = g(q; p) 1) A major disadvantage of the general Poincar e 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 Poincar e transformation is that it allows the ....

E. Hairer, Variable time step integration with symplectic methods, Applied Numerical Mathematics, 25 (1997), pp. 219--227.

....explains 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 ....

E. Hairer, Variable Timestep Integration with Symplectic Methods, Appl. Numer. Math. 25, 219-227, 1997.

....of variable stepsizes destroys the favorable longtime behavior. In some situations (e.g. planetary orbits with large eccentricity) the use of variable stepsizes is indispensable for an efficient integration. There, it is possible to reparametrize time [19, 25, 16, 18] or to scale the Hamiltonian [9, 22] so that the use of constant stepsizes for the new problem corresponds to a variable stepsize integration of the original one. 3. Multistep methods. It is well known that multistep methods have many advantages when constant stepsizes are used. They can be implemented very efficiently, and it is ....

E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math. 25 (1997), pp. 219--227.

....of variable stepsizes destroys the favorable longtime behavior. In some situations (e.g. planetary orbits with large eccentricity) the use of variable stepsizes is indispensable for an efficient integration. There, it is possible to reparametrize time [18, 23, 15, 17] or to scale the Hamiltonian [8, 20] so that the use of constant stepsizes for the new problem corresponds to a variable stepsize integration of the original one. 3. Multistep methods. It is well known that multistep methods have many advantages when constant stepsizes are used. They can be implemented very efficiently, and it is ....

E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math. 25 (1997), pp. 219--227.

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E. Hairer, Variable Time Step Integration with Symplectic Methods, Appl. Num. Math. 25, 1997, 219-227.

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E. Hairer, Variable time step integration with symplectic methods, Appl. Numer. Math., 25 (1997), pp. 219-227.

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