Variable stepsize

Robert D. Skeel and Jeffrey J. Biesiadecki. Symplectic integration with variable stepsize. Ann. Num. Math., 1 191-198, 1994. 

Gradient calculations for the x variables are obtained from implicit reformulations of the DAE model. Clearly the easiest, but least accurate, way is simply to re-solve the model for each perturbation of the parameters. Sargent and Sullivan (1977, 1979) derived these gradients using an adjoint formulation. In addition, they were able to accelerate the adjoint computations by retaining the information from the model solution (the forward step) for the adjoint solution in the backward step. This approach was later refined for variable stepsize methods by Morison (1984). The adjoint approach to parameterized optimal control was also used by Jang et al. (1987) and Goh and Teo (1988). 

Barth, E., Leimkuhler, B., Reich, S. Time-reversible variable-stepsize integrator for constrained dynamics. SIAM J. Sci. Comput. 1999, 21,1027-44. 

A variable stepsize is only used where one expects extreme chtinges in particle velocities over the course of a simulation (see e.g. [75,390] for examples arising in radiation damage cascades). 

For our chemical problem, we successfully use the stiff versions of the solvers LSODE (from A.C.Hindmarsh) and VODE (from G.D.Byrne and A.C.Hindmarsh), which employ multistep methods (backward differentiation formulas) and allow to change stepsize and order of the methods. Comparing investigations show that the IVP-solver RODAS (from E.Hairer and G.Wanner), an implementation of onestep methods (Rosenbrock methods), also with variable stepsize, is working with same success (see e.g. [4, 5]). 

Higinio Ramos and Jesus Vigo-Aguiar, Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P. s, J. Comput. Appl. Math.,... 

In ref 139 the authors presented variable-stepsize Chebyshev-type methods for the integration of second-order initial-value problems. More specifically, Panovsky and Richardson in ref. 140 presented a method based on Chebyshev approximations for the numerical solution of the problem y" = f(x,y), with constant stepsize. In ref. 141 Coleman and Booth analyzed the method developed in ref 140 and proposed the convenience to design a variable stepzesize methods of Chebyshev-type. The development of the new methods is based on the test equation ... 

In ref. 161 the authors develop numerical methods based on Chebyshev approximations with variable stepsize. This was an extension of the papers of J. Panovsky and D. L. Richardson and of John P. Coleman, Andrew S. Booth. ... 

The one-step methods described above are the Taylor series approximation where the higher derivative terms are ignored. However, many times this approximation may not hold true, and variable stepsize methods are then preferable. In spaces where higher-order derivatives cannot be ignored for that step, use very small h, otherwise, use a larger step size. Several different variants of the methods presented here are available where higher order derivative information can be used to obtain automatic step size changes. For more details, refer to [9]-[10]. ... 

Rwork(41) contains the relative perturbation step size for finite-difference computation of G(t). The default stepsize is the square root of the machine precision DDAPLUS uses this unless MAIN provides a larger value. Rwork(42) must be set to the desired stopping value Cstop of the state or sensitivity variable U(IYstop). whenever the user inserts an index lYstop into Info (17). 

Table 13 RTC (real of computation (in seconds)) to calculate S 2 for the variable-step methods (l)-(20). acc = 10 6. hmax is the maximum stepsize...

Figure 24 Period-doubling bifurcations in the Rossier model, Eqs. [106]. For these simulations (computed using the Runge-Kutta routine DiffEq-3D, with stepsize = 0.1, available through Ref. 18), the following parameters were used a = 0.1 and b = 0.2 with c variable. In (a) c = 2.5 ind the initial values were Xq = -2.98, =... 

A step can then be taken to the first point of subsequent collision, with positions updated using the quadratic and momenta adjusted according to the Verlet map combined with In this way, all steps taken are Verlet steps so the order of accuracy is two. Effectively, this is a Verlet method with variable timestep chosen to match collision times. A maximum outer stepsize fimax is incorporated to prevent too-large steps from being taken (in case collisions are infrequent). We write the method here in a slightly different form than in [184] both the stepsize taken and the coordinates are viewed as variables. 

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