nLab Runge-Kutta method

Runge-Kutta approximation schemes are a family of difference schemes used for iterative numerical solution of ordinary differential equations.

Standard Runge-Kutta methods

In engineering it is rather standard to use the 4th order Runge-Kutta difference schemes. General overview is at

• wikipedia Runge-Kutta methods

• H. Munthe-Kaas, High order Runge–Kutta methods on manifolds, Appl. Num. Math., 29, 115-127 (1999)

For stochastic ODE-s:

• wikipedia Runge–Kutta method (SDE)

Relation to algebraic problems, Lie theory and renormalization

• J.C. Butcher, Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc. 3 (1963) 185–201

In the 1972 Butcher’s work, Runge-Kutta methods are organized into a group, later called Butcher group:

• J. C. Butcher, An algebraic theory of integration methods, Math. Comp. 26 (1972) 79–106.

Brouder has shown a relation of Butcher group to Connes-Kreimer Hopf algebra.

• Ch. Brouder, Runge-Kutta methods and renormalization, Europ. Phys. J. C12 (2000) 512–534.

The Butcher-Connes-Kreimer Hopf algebra is a Hopf subalgebra or Foissy Hopf algebra from work

• L. Foissy, Les algèbres de Hopf des arbres enracinés décorés, I., Bulletin des Sciences Mathematiques 126, no. 3 (2002): 193–239 (arXiv:math/0105212). MR1905177 Les algèbres de Hopf des arbres enracinés décorés. II. Bulletin des Sciences Mathematiques 126, no. 4 (2002): 249–288 (arXiv:math/0105212) MR1909461

• Hans Z. Munthe-Kaas, Ari Stern, Olivier Verdier, Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds, SIAM Journal on Applied Algebra and Geometry 4 (1) 49–68 (2020) doi arXiv:1903.10056

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numerical integrators, giving a characterization of the spaces on which Butcher and Lie–Butcher series methods, which generalize Runge–Kutta methods, may be applied.

Important family of Lie-algebraic methods for generating integrators are introduced in

• P. E. Crouch, R. Grossman, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci. 3: 1–33 (1993) doi

Connections to symplectic geometry

See also symplectic integrators.

• Alberto S. Cattaneo, Benoit Dherin, Giovanni Felder, Formal symplectic groupoid, Comm. Math. Phys. 253 (2005), no. 3, 645–674 math.SG/0312380 doi

The multiplicative structure of the trivial symplectic groupoid over $\mathbb{R}^d$ associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it. This formula turns out to be the semi-classical approximation of Kontsevich’s deformation formula. For the case of a linear Poisson structure, the deformed generating function reduces exactly to the CBH formula of the associated Lie algebra. The methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics: the Runge–Kutta theory of the numeric integration of ODE’s.

• Robert McLachlan, Klas Modin, Olivier Verdier, Collective Lie–Poisson integrators on $\mathbb{R}^{3}$, arxiv/1307.2387

We develop Lie–Poisson integrators for general Hamiltonian systems on $\mathbb{R}^{3}$ equipped with the rigid body bracket. The method uses symplectic realisation of $\mathbb{R}^{3}$ on $T^{*}\mathbb{R}^{2}$ and application of symplectic Runge–Kutta schemes. As a side product, we obtain simple symplectic integrators for general Hamiltonian systems on the sphere $S^{2}$.

• J. M. Sanz-Serra, Runge-Kutta Schemes for Hamiltonian Systems, Bit 28, pp. 877-883, (1988).