In this work, the benefits of the phase fitting technique are embedded in high order discrete Lagrangian integrators. The proposed methodology creates integrators with zero phase lag in a test Lagrangian in a similar way used in phase fitted numerical methods for ordinary differential equations. Moreover, an efficient method for frequency evaluation is proposed based on the eccentricities of the moving objects. The results show that the new method dramatically improves the accuracy and total energy behaviour in Hamiltonian systems. Numerical tests for the 2-body problem with ultra high eccentricity up to 0.99 for 1000000 periods and to the Henon-Heiles Hamiltonian system with chaotic behaviour, show the efficiency of the proposed approach.
Deep Dive into High Order Phase-fitted Discrete Lagrangian Integrators for Orbital Problems.
In this work, the benefits of the phase fitting technique are embedded in high order discrete Lagrangian integrators. The proposed methodology creates integrators with zero phase lag in a test Lagrangian in a similar way used in phase fitted numerical methods for ordinary differential equations. Moreover, an efficient method for frequency evaluation is proposed based on the eccentricities of the moving objects. The results show that the new method dramatically improves the accuracy and total energy behaviour in Hamiltonian systems. Numerical tests for the 2-body problem with ultra high eccentricity up to 0.99 for 1000000 periods and to the Henon-Heiles Hamiltonian system with chaotic behaviour, show the efficiency of the proposed approach.
In the field of numerical integration, methods specially tuned on oscillating functions, are of great practical importance. Such methods are needed in various branches of natural sciences, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behaviour. For a review of such methods see Ixaru et al. (1997);VandenBerghe et al. (1999VandenBerghe et al. ( , 2001)); Ixaru et al. (2003); VanDaele and VandenBerghe (2007) and references there in as well as the book Ixaru and VandenBerghe (2004).
For problems having highly oscillatory solutions, standard methods with unspecialized use can require a huge number of steps to track the oscillations. One way to obtain a more efficient integration process is to construct numerical methods with an increased algebraic order, although the simple implementation of high algebraic order methods may cause several problems (for example, the existence of parasitic solutions Quinlan (1999)). On the other hand, there are some special techniques for optimizing numerical methods. Trigonometrical fitting and phase-fitting are some of them, producing methods with variable coefficients, which depend on v = ωh, where ω is the dominant frequency of the problem and h is the step length of integration. This technique is known as exponential (or trigonometric if µ = iω) fitting and has a long history Gautschi (1961), Lyche (1972). An important property of exponential fitted algorithms is that they tend to the classical ones when the involved frequencies tend to zero, a fact which allows to say that exponential fitting represents a natural extension of the classical polynomial fitting. The examination of the convergence of exponential fitted multistep methods is included in Lyches theory Lyche (1972). The general theory is presented in detail in Ixaru and VandenBerghe (2004). Furthermore, considering the accuracy of a method when solving oscillatory problems, it is more appropriate to work with the phase-lag, rather than its usual primary local truncation error. We mention the pioneering paper of Brusa and Nigro Brusa and Nigro (1980), in which the phase-lag property was introduced. This is actually another type of a truncation error, i.e. the angle between the analytical solution and the numerical solution. A significant application of the phase or exponential fitting is on the construction of symplectic methods for oscillatory problems encountered in physics and chemistry (Monovasilis et al. (2005(Monovasilis et al. ( , 2006))).
Although phase fitting and exponential fitting are a major improvement over algebraic fitted methods especially for oscillatory and orbital problems, there is not significant evidence from published results that these methods can be applied for long term integration (for example for millions or billions of periods). Moreover, several authors use to test their methods to the well known 2-body problem but only for relatively low eccentricities (up to 0.2) and for relatively small number of periods (no more than several thousands). We mention here the efforts of VandeVyver (2006VandeVyver ( , 2005)); Wang (2005);Simos (2004); Anastassi andSimos (2005, 2004) in which there is no evidence that the phase fitting or trigonometric fitting can be applied to high eccentricities (for example to the Halley comet with eccentricity close to 0.967) and for long time.
Another approach to oscillatory and especially Hamiltonian systems is the theory of discrete variational mechanics, which was set up in the 1960s Jordan and Polak (1964); Cadzow (1970); Logan (1973) and then it was proposed in the optimal control literature. It then motivated a lot of authors and soon the discrete Euler-Lagrange equations were formulated and the first integrators in the discrete calculus of variation and further the multi-freedom and higher-order problems were studied. Afterwards, the canonical structure and symmetries for discrete systems were obtained, and Noether’s theorem to the discrete case was extended Maeda (1980Maeda ( , 1981)). Finally, the time as a discrete dynamical variable was regarded Lee (1983). A detailed description of the essential properties of variational integrators can be found in Marsden and West (2001); Marsden et al. (1998); Lee (1983). One of the most important properties of variational integrators is that since the discrete Lagrangian is an approximation of a continuous Lagrangian function, the obtained numerical integrator inherits some of the geometric properties of the continuous Lagrangian (such as symplecticity, momentum preservation).
In the present work, the benefits of the two approach are combined in order to construct high order discrete Lagrangian integrators with phase fitting. To obtain this, we have adopted a test Lagrangian problem (similar to the test ODE in the phase fitting) which is the harmonic oscillator with given frequency ω. Then, we construct discrete variational schemes that solve exactly the test Lagrangia
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