gautschiIntegrators

A package for integrating oscillating second order differential equations $$x'' = -Ax + g(x).$$

The integrators are described in [2]. They make use of trigonometric functions to take advantage of known structure of the solution, as proposed by [4], after whom this package is named.

The matrix functions appearing in the integrators can be evaluated by symmetric diagonalization or by using PyWkm, which is a private package at the moment. Both of these methods can also be combined with the (restarted) Lanczos-method for evaluating matrix functions times a vector.

Usage

See examples/ for example usage of the integrators for a Fermi-Pasta-Ulam-Tsingou problem.

The central function is solve_ivp in integrate.py. It is modeled after the Scipy function of the same name, but has differing in- and outputs.

See the METHODS dictionary in integrate.py for the available one- and two-step configurations of the gautschiIntegrators.

The matrix functions are evaluated by an instance of the MatrixFunctionEvaluator base class. Subclasses need to implement the calculation of $\cos(\sqrt(A))b$, $\sinc(\sqrt(A))b$ and $\sqrt(A)\sin(\sqrt(A))b$. This can be achieved for example by diagonalization of symmetric tri-diagonal matrices as in TridiagDiagonalizationEvaluator.

Requirements

The code of PyWkm needs to be copied into a folder pywkm for usage in this repository.

This package does not use low-level methods from its dependencies, so I would expect it to work with any current Numpy and Scipy versions. environment.yml gives specifications for a Conda environment that this package has been used with.

Examples

Gautschi-type integrators have usually been evaluated on test problems inspired by the Fermi-Pasta-Ulam-Tsingou experiment, e.g., in [2]. FermiPastaUlamTsingou.py contains a single frequency example from Section XIII.2.1 of [3]. FPUTMultiFrequency.py contains a multi frequency example from [1]. FPUTMultiFrequencyLanczos.py is a variation of the former with longer vectors, so that the Lanczos method can be used to approximate the matrix functions.

Notes

The git branch titled fermi-pasta-lanczos-experiment contains the multi-frequency Fermi-Pasta-Ulam-Tsingou problem inspired by [1] in FPUTMultiFrequencyLanczosLonger.py. It is similar to FPUTMultiFrequencyLanczos.py, but simulates a longer time.

References

[1] D. Cohen, E. Hairer, and Ch. Lubich, “Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations,” Bit Numer Math, vol. 45, no. 2, pp. 287–305, Jun. 2005, doi: 10.1007/s10543-005-7121-z.

[2] E. Hairer and C. Lubich, “Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations,” SIAM J. Numer. Anal., vol. 38, no. 2, pp. 414–441, Jul. 2000, doi: 10.1137/S0036142999353594.

[3] E. Hairer, G. Wanner, and C. Lubich, Geometric Numerical Integration, vol. 31. in Springer Series in Computational Mathematics, vol. 31. Berlin/Heidelberg: Springer-Verlag, 2006. doi: 10.1007/3-540-30666-8.

[4] W. Gautschi, “Numerical integration of ordinary differential equations based on trigonometric polynomials,” Numerische Mathematik, vol. 3, no. 1, pp. 381–397, Dec. 1961, doi: 10.1007/BF01386037.