Bibliography

[1]
C. Rackauckas and Q. Nie. DifferentialEquations.jl - A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia. Journal of Open Research Software 5, 15 (2017).
[2]
J. Bezanson, A. Edelman, S. Karpinski and V. B. Shah. Julia: A Fresh Approach to Numerical Computing. Siam Review 59, 65–98 (2017).
[3]
W. Fangzong and L. Xiaobing. A Class of Lobatto Methods of Order 2s. International Journal of Applied Mathematics 46, 6–10 (2016).
[4]
R. I. McLachlan and G. R. Quispel. Splitting methods. Acta Numerica 11, 341–434 (2002).
[5]
E. Hairer, C. Lubich and G. Wanner. Geometric Numerical Integration (Springer, 2006).
[6]
G. Strang. On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis 5, 506–517 (1968).
[7]
G. I. Marchuk. Some applications of splitting-up methods to the solution of mathematical physics problems. Aplikace Matematiky 13, 103–132 (1968).
[8]
R. I. McLachlan. On the Numerical Integration of Ordinary Differential Equations by Symmetric Composition Methods. SIAM Journal on Scientific Computing 16, 151–168 (1995).
[9]
J. E. Marsden and M. West. Discrete Mechanics and Variational Integrators. Acta Numerica 10, 357–514 (2001).
[10]
A. P. Veselov. Integrable discrete-time systems and difference operators. Functional Analysis and Its Applications 22, 83–93 (1988).
[11]
J. M. Wendlandt and J. E. Marsden. Mechanical integrators derived from a discrete variational principle. Physica D: Nonlinear Phenomena 106, 223–246 (1997).
[12]
J. E. Marsden and J. M. Wendlandt. Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms. In: Current and Future Directions in Applied Mathematics, edited by M. Alber, B. Hu and J. Rosenthal (Birkhäuser Boston, 1997); pp. 219–261.
[13]
M. Leok and J. Zhang. Discrete Hamiltonian Variational Integrators. IMA Journal of Numerical Analysis 31, 1497–1532 (2011).
[14]
M. Kraus. Projected Variational Integrators for Degenerate Lagrangian Systems, arXiv:1708.07356.
[15]
S. Ober-Blöbaum. Galerkin variational integrators and modified symplectic Runge-Kutta methods. IMA Journal of Numerical Analysis 37, 375–406 (2016).
[16]
T. Dray. Differential Forms and the Geometry of General Relativity (CRC Press, 2014).
[17]
J. Baez and J. P. Muniain. Gauge Fields, Knots and Gravity (World Scientific, 1994).
[18]
R. W. Darling. Differential Forms and Connections (Cambridge University Press, 1994).
[19]
T. Frankel. The Geometry of Physics (Cambridge University Press, 2011).
[20]
J. M. Lee. Introduction to Smooth Manifolds (Springer, 2012).
[21]
J. M. Lee. Manifolds and Differential Geometry (American Mathematical Society, 2009).
[22]
L. W. Tu. An Introduction to Manifolds (Springer, 2011).
[23]
S. Morita. Geometry of Differential Forms (American Mathematical Society, 2001).
[24]
R. Abraham and J. E. Marsden. Foundations of Mechanics (American Mathematical Society, 1978).
[25]
J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry (Springer, 2002).
[26]
D. D. Holm, T. Schmah and C. Stoica. Mechanics and Symmetry (Oxford University Press, 2009).
[27]
E. Hairer. Geometric Integration of Ordinary Differential Equations on Manifolds. BIT Numerical Mathematics 41, 996–1007 (2001).
[28]
E. Hairer. Symmetric Projection Methods for Differential Equations on Manifolds. BIT Numerical Mathematics 40, 726–734 (2000).
[29]
R. P. Chan, P. Chartier and A. Murua. Reversible methods of Runge-Kutta type for Index-2 DAEs. Numerische Mathematik 97, 427–440 (2004).
[30]
E. Hairer, C. Lubich and M. Roche. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Vol. 1409 of Lecture Notes in Mathematics (Springer, 1989).
[31]
U. M. Ascher and L. R. Petzold. Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM Journal on Numerical Analysis 28, 1097–1120 (1991).
[32]
R. P. Chan, P. Chartier and A. Murua. Post-projected Runge-Kutta methods for index-2 differential-algebraic equations. Applied Numerical Mathematics 42, 77–94 (2002).
[33]
L. O. Jay. Solution of Index 2 Implicit Differential-Algebraic Equations by Lobatto Runge-Kutta Methods. BIT Numerical Mathematics 43, 93–106 (2003).
[34]
L. O. Jay. Specialized Runge-Kutta methods for index 2 differential-algebraic equations. Mathematics of Computation 75, 641–654 (2006).
[35]
W. Oevel and M. Sofroniou. Symplectic Runge-Kutta schemes II: classification of symmetric methods. Preprint.
[36]
S. Zhao and G.-W. Wei. A unified discontinuous Galerkin framework for time integration. Mathematical Methods in the Applied Sciences 37, 1042–1071 (2014).
[37]
L. O. Jay. Structure Preservation for Constrained Dynamics with Super Partitioned Additive Runge-Kutta Methods. SIAM Journal on Scientific Computing 20, 416–446 (1998).
[38]
L. O. Jay. Specialized Partitioned Additive Runge-Kutta Methods for Systems of Overdetermined DAEs with Holonomic Constraints. SIAM Journal on Numerical Analysis 45, 1814–1842 (2007).
[39]
L. O. Jay. Beyond conventional Runge-Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models. Journal of Computational and Applied Mathematics 204, 56–76 (2007).
[40]
M. Kraus. SPARK Methods for Degenerate Lagrangian Systems. In preparation.
[41]
M. Leok. Generalized Galerkin Variational Integrators: Lie group, multiscale, and pseudospectral methods, arXiv:math.NA/0508360.
[42]
C. M. Campos. High Order Variational Integrators: A Polynomial Approach. In: Advances in Differential Equations and Applications (Springer International Publishing, 2014); pp. 249–258.
[43]
C. M. Campos, S. Ober-Blöbaum and E. Trélat. High order variational integrators in the optimal control of mechanical systems. Discrete and Continuous Dynamical Systems 35, 4193–4223 (2015).
[44]
S. Ober-Blöbaum and N. Saake. Construction and analysis of higher order Galerkin variational integrators. Advances in Computational Mathematics 41, 955–986 (2014).