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Commutator-free Lie group methods with minimum storage requirements and reuse of exponentials

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Abstract

A new format for commutator-free Lie group methods is proposed based on explicit classical Runge-Kutta schemes. In this format exponentials are reused at every stage and the storage is required only for two quantities: the right hand side of the differential equation evaluated at a given Runge-Kutta stage and the function value updated at the same stage. The next stage of the scheme is able to overwrite these values. The result is proven for a 3-stage third order method and a conjecture for higher order methods is formulated. Five numerical examples are provided in support of the conjecture. This new class of structure-preserving integrators has a wide variety of applications for numerically solving differential equations on manifolds.

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Notes

  1. It is implied here and later on that when the upper bound on the index in a sum is smaller than the lower bound, the sum is set to 0 and if the same conditions hold for a product, the product is set to 1, e.g. \(\sum _{j=1}^0...=0\), \(\prod _{j=1}^0...=1\).

  2. by analogy with quantum field theory

  3. The clash of notation here is unfortunate, but it is customary in the literature on low-storage schemes to use \(A_i\) for the coefficients, while it is customary in the literature on Lie group methods to use A(Y) on the right hand side of Eq. (3.1).

  4. and all 3-stage third-order explicit RK schemes satisfying the constraint (4.17)

  5. up to the dimension: we use \(5\times 5\) orthogonal matrices and Ref. [23] used \(4\times 4\)

  6. The simplification that allows for this observation can be traced to the fact that \(b_2=0\).

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Acknowledgements

I thank Andrea Shindler for careful reading and comments on the manuscript and Oswald Knoth for bringing my attention to Ref. [18, 31], comments on the manuscript and, most importantly, for independently checking the order conditions for all coefficient schemes of Table 1 (as Lie group integrators of the form (4.18)–(4.19)) with his software available at [17] and explaining me the theory and algorithmic details behind that software. This work was in part supported by the U.S. National Science Foundation under award PHY-1812332.

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  1. Department of Computational Mathematics, Science and Engineering, Department of Physics and Astronomy, Michigan State University, East Lansing, MI, 48824, USA

    Alexei Bazavov

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Correspondence to Alexei Bazavov.

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Communicated by Antonella Zanna Munthe-Kaas.

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Matlab code

Matlab code

To illustrate the usage of the 2N-storage schemes as commutator-free Lie group integrators we present below a Matlab script that generates a scaling plot similar to Fig. 3 using the integrators BWRRK33, TSRKF84 and YRK135 from Table 1.

figure a
figure b
figure c
figure d

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Bazavov, A. Commutator-free Lie group methods with minimum storage requirements and reuse of exponentials. Bit Numer Math 62, 745–771 (2022). https://doi.org/10.1007/s10543-021-00892-x

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