Extensions of B-series

[ketch]

There are several extensions of "B-series for Runge-Kutta methods" that could also be implemented in this package:

• P-series for partitioned methods (see HNW II.15) and NB-series for additive methods (see e.g. https://www.sciencedirect.com/science/article/pii/S0168927498000646)
  • Basic implementation via AdditiveRungeKuttaMethods in time integration methods and multirate infinitesimal split methods #43
  • Backward error analysis stuff (substitute needed for modified_equation, modifying_integrator) (implemented in substitute and backward error analysis for colored trees / additive RK methods #51 and More backward error analysis with additive decompositions #53)
  • compose
  • latexify
• S-series (see again paper above)
• Runge-Kutta-Nyström methods
  • Can basically be interpreted as P-series with a subset of all order conditions due to the special form (u, u')' = (u', f)
• Generalized additive RK (GARK) methods
• LS-series
• B-Series for multiderivative methods (HNW II.13)
• B-series of two-derivative Runge-Kutta methods
  • https://doi.org/10.1007/s11075-009-9349-1
• B-Series for general linear methods (HNW III.8; Butcher B-Series Ch. 6)
• B-series for Rosenbrock methods (HNW exercise IV.7.2)
• B-series for Rosenbrock W methods
• B-series for exponential integrators
  • https://epubs.siam.org/doi/pdf/10.1137/040612683
  • Vu Thai Luan, Trky Alhsmy (2023) Derivation of sixth-order exponential Runge--Kutta methods for stiff systems
• B-series for Elementary Differential Runge-Kutta methods
  • Sofroniou, Mark, and Giulia Spaletta. "On the construction of a new generalization of Runge–Kutta methods." Electronic Notes in Theoretical Computer Science 74 (2003): 189-206. https://doi.org/10.1016/S1571-0661(04)80774-6
• Stochastic differential equations (e.g., Rößler, https://doi.org/10.1137/09076636X)
• Application to effective order methods (using the composition law)
• DAE-series for differential-algebraic equations
• Lie-Butcher series
• Elementary Hamiltonians and modified Hamiltonian (Section IX.9.2 of Hairer, Lubich, Wanner (2006) Geometric numerical integration, pp. 373ff)
• aromatic B-series
  • Munthe-Kaas, Verdier: Aromatic Butcher Series, https://doi.org/10.1007/s10208-015-9245-0
  • Bogfjellmo: Algebraic structure of aromatic B-series, http://dx.doi.org/10.3934/jcd.2019010
• Exotic B-series and S-series
  • Bronasco, https://arxiv.org/abs/2209.11046

We should also extend the analysis and handling of B-series, e.g.,

• order_of_accuracy
• inverse (with respect to compose)
• compute the adjoint method
• checking symmetry
• checking Hamiltonian structure (of vector fields)
• checking symplectic structure:
  • single-colored trees order_of_symplecticity and is_symplectic #216
  • bi-colored trees for partitioned systems
• checking energy preservation
• checking the properties above up to conjugation

A nice test of pseudo-symmetry and pseudo-symplecticity could be the complex compisitions of Casas, Chartier, Escorihuela-Tomas, and Zhang (2021), https://doi.org/10.1016/j.cam.2020.113006

[ranocha]

What would be necessary to support these extensions and how could we approach that? As far as I know, we would need another kind of rooted trees with (at least) two different leaf types for some of these extensions, correct? Is there something "cheaper" we could look at first that doesn't require other tree types? That might help us to improve the interfaces etc. we're using.

[ketch]

I don't know completely what is needed, but yes, one of the main things would be to support trees with two types of nodes (or n types of nodes for n-additive methods).

[dynamic-queries]

@ketch @ranocha Could you suggest a good first issue?

[ketch]

@dynamic-queries Thanks for the interest; it depends quite a bit on your background. Are you already familiar with B-series (as a mathematical tool)? With time integration methods?

[dynamic-queries]

@dynamic-queries Thanks for the interest; it depends quite a bit on your background. Are you already familiar with B-series (as a mathematical tool)? With time integration methods?

@ketch I am a masters student in scientific computing with some experience in time integration methods.
B-series is new-territory for me. I am reading Algebraic Structures of B-series to start with.

[ranocha]

Thanks for your interest! If you're new to Julia and/or GitHub, you could start with a small PR, e.g., #103
If you are interested in time integration methods and want to implement something completely new, you could check whether the class of time integration schemes you are interested in is already covered by BSeries.jl. If not, you can implement it.
Something a bit simpler would be to implement new functions checking properties of B-series, e.g., symplecticity. The book "Geometric Numerical Integration" of Hairer et al. describes the theory in Section VI.7.