S. Bond and B. Leimkuhler, Time-transformations for reversible variable stepsize integration, Numerical Algorithms, (1998), pp. 55-71.  Home/Search   Document Details and Download   Summary   Related Articles   Check

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S. Bond and B. Leimkuhler, Time-transformations for reversible variable stepsize integration, Numerical Algorithms, (1998), pp. 55-71.

.... desired, a third order splitting method could be used in place of t;H , resulting in a fourth order method overall (see [19] for a discussion of how to construct such splittings) We anticipate that the considerations for the choice of time transformation will be similar to those discussed in [2, 15]. In the latter reference, it was 6 anne kv rn and ben leimkuhler suggested to use a control of the form g min = 1 1 r 3=2 (3) where r represents the smallest pair separation. In some cases, it may be desirable to have a smooth control, in which case we could use g sm = 1 1 P ....

....ben leimkuhler suggested to use a control of the form g min = 1 1 r 3=2 (3) where r represents the smallest pair separation. In some cases, it may be desirable to have a smooth control, in which case we could use g sm = 1 1 P i j r 3m=2 ij 1=m ; where m is a positive integer [2]) 3. Integration of H loc . In this and the following section, we describe the integrator for H loc . We will assume that the solution is desired on the time interval [0; t] that initial positions q 0 and momenta p 0 are provided and that new values q 1 and p 1 are to be computed. ....

Bond, S., and Leimkuhler, B., Time-transformations for reversible variable stepsize integration, Numerical Algorithms. 19, 55-71, 1998.

.... a third order splitting method could be used in place of # # #### , resulting in a fourth order method overall (see [19] for a discussion of how to construct such splittings) Weanticipate that the considerations for the choice of time transformation will be similar to those discussed in [2, 15]. In the latter reference, it was 6 anne kv rn and ben leimkuhler suggested to use a control of the form # ### = 1 1 # #### (3) where # represents the smallest pair separation. In some cases, it may be desirable to have a smooth control, in which case we could use # ## = 1 1 P ### # ....

....and ben leimkuhler suggested to use a control of the form # ### = 1 1 # #### (3) where # represents the smallest pair separation. In some cases, it may be desirable to have a smooth control, in which case we could use # ## = 1 1 P ### # ##### ## ### # where # is a positiveinteger [2]) 3. Integration of # ### . In this and the following section, we describe the integrator for # ### . We will assume that the solution is desired on the time interval [0# #] that initial positions # # and momenta # # are provided and that new values # # and # # are to be computed. ....

Bond, S., and Leimkuhler, B., Time-transformations for reversible variable stepsize integration, ######### ##########. ##, 55-71, 1998.

....interactions are not treated with any regularization other than the time transformation itself. We can gain insight into this term by considering the purely repulsive 19 one degree of freedom problem with Hamiltonian H = 1 2 p 2 1 q ; q 0: 6. 25) We follow the approach given in [5] to obtain an appropriate time transformation for solving this model problem. After a time rescaling g = q fi , the equations of motion for (6.25) are q = q fi p p = q fi Gamma2 Along an orbit, the material point starts from a distant point (momentum p Gamma1 ) and approaches the ....

....r ee in the control. Although the optimal value in this experiment occurs at around b = 2 a discrepancy with our theoretical discussion the results for b = 1:5 are similar. The resulting energy error and stepsize graphs are shown in Fig. 10. Additional discussion and examples may be found in [5]. 20 1 0.5 0 0.5 1 1 0.5 0 0.5 1 x 1 0.5 0 0.5 1 1 0.5 0 0.5 1 x Figure 8: The first test orbit for Helium, near a Langmuir orbit, for which the timestep selection is dominated by electron electron interactions. Left, to T=30; right, to T=300. 6.1.2 Electron nucleus ....

S. Bond and B. Leimkuhler, Time transformations for reversible variable stepsize integration, preprint, submitted for publication, 1998.

....vg v : 2.16) If we set r = p x 2 y 2 and suppose that g g(r) then (2.16) reduces to g = 2 3 rg r ; so that g = r 3=2 : 2.17) We then have x = r 3=2 u; u = x=r 3=2 ; y = r 3=2 v; v = y=r 3=2 ; t = r 3=2 : 2. 18) The discretisation of this system was studied in [3]. To summarise, the proposed invariant Sundman rescaling of (1.1) is given by du i =d = g(u)f i (u) dt=d = g(u) 2.19) where g is any function satisfying the hyperbolic equation (2.13) such that g( 1 u 1 ; 2 u 2 ; f i ( 1 u 1 ; 2 u 2 ; i g(u)f i (u) 2.20) ....

S.D. Bond and B.J. Leimkuhler, \Time-Transformations for Reversible Variable Stepsize Integration", Numerical Algorithms, 19, (1998), pp. 55-71.

.... original time variable t is recovered from the update t n 1 = t n Deltas 2 1 ae n 1 ae n 1 which is a discretization of dt ds = 1 U(x) Other choices of of the scaling function have been developed in the context of integration of few body systems with inverse power potentials [2]. Note that the suggested symmetric, variable stepsize method (19) 21) is not restricted to secondorder methods. Often higher order methods for time reversible differential equations are based on an appropriate concatenation of a symmetric second order method. For example, let Phi SV Deltat ....

S. Bond and B. Leimkuhler, Time-transformations for reversible variable stepsize integration, Numerical Algorithms, to appear.

.... a third order splitting method could be used in place of Phi Deltat;H , resulting in a fourth order method overall (see [17] for a discussion of how to construct such splittings) We anticipate that the considerations for the choice of time transformation will be similar to those discussed in [2, 13]. In the latter reference, it was suggested to use a control of the form g min = 1 1 r Gamma3=2 where r represents the smallest pair separation. In some cases, it may be desirable to have a smooth control, in which case we could use g sm = 1 1 i P i j r Gamma3m=2 ij j 1=m ; ....

....it was suggested to use a control of the form g min = 1 1 r Gamma3=2 where r represents the smallest pair separation. In some cases, it may be desirable to have a smooth control, in which case we could use g sm = 1 1 i P i j r Gamma3m=2 ij j 1=m ; where m is a positive integer [2]) DESIGN OF A MECHANICAL N BODY INTEGRATOR 7 3. Integration of H loc . We now describe the integrator for H loc , a key component of the overall scheme. We will assume throughout this section that the solution is desired on the time interval [0; Deltat] that initial positions q 0 and ....

Bond, S., and Leimkuhler, B., Time-transformations for reversible variable stepsize integration, Numerical Algorithms, to appear.

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S.D. Bond and B. Leimkuhler, Time-transformations for reversible variable step-size integration, Numerical Algorithms, 19 (1998), pp. 55-71.

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