arXiv subjects

Eugen Bronasco

Publications and source records attributed to Eugen Bronasco.

8 recordsLinked to original sources

arXiv2026

Convergence analysis of generalized modified splitting methods using multi-index series

We consider splitting methods for partial differential equations involving unbounded operators. For non-time-reversible dynamics, such as dissipative systems, negative splitting coefficients are generally not admissible because they require stepping backward in time, leading to an order barrier when all coefficients are required to be positive. We introduce generalized modified splitting methods to overcome this barrier. To analyze their convergence, we develop the corresponding multi-index series formalism, which provides a systematic framework for deriving order conditions. Using this formalism, we derive the order conditions and construct a generalized modified splitting method of order $6$. We also provide Python scripts that automate the generation and verification of order conditions, as well as the construction of new generalized modified splitting methods. Finally, we establish connections between the introduced multi-index series and related series formalisms from the literature, in particular, word series, Lie-Butcher series, and multi-index Butcher series.

math.NA↗

arXiv2026

Backward error analysis for matrix discretizations of 2-D Euler equations

We introduce a formalism of Lie--Poisson reduction of Butcher series. The corresponding forest momentum map allows for describing backward error analysis of isospectral symplectic Runge--Kutta methods applied to Zeitlin's matrix discretization of the 2-D Euler equations on the sphere. Based thereon, we obtain exponentially small error bounds for the conservation of modified Hamiltonians, valid for exponentially long time intervals. Crucially, the error bounds and the length of the time intervals are independent of the spatial discretization parameter $n$ (the matrix size) when the time step for different $n$ is scaled as $h = \mathcal{O}(n^{-1})$. Our results thus extend the classical backward error analysis result for finite-dimensional Hamiltonian systems to the infinite-dimensional case of the 2-D Euler equations discretized via matrix hydrodynamics.

math.NA↗

arXiv2026

Arboretum.hs: Symbolic manipulation for algebras of graphs

We design the Arboretum$.$hs package for symbolic computations with algebras of trees and more general graphs in Haskell. Thanks to the declarative nature of functional programming, the package's implementation closely follows mathematical definitions, making the code intuitive and transparent for users working with algebraic and combinatorial structures. To assist with current mathematical research, Arboretum$.$hs supports experimentation by facilitating the introduction of new algebraic operations, as well as providing functionality for rendering trees and forests through LaTeX integration. Compared to recent imperative implementations in languages such as Julia or Python, Arboretum$.$hs offers greater flexibility for manipulating and extending tree-based structures. Its use of Haskell enables safe programming and strong compile-time guarantees, serving both as a practical computational tool and a foundation for further research in algebraic combinatorics, beyond the setting of trees usually considered in the implementation of Butcher series, which are a fundamental tool for the analysis of numerical integrators.

cs.SC↗

arXiv2026

Spectral Deferred Corrections in the framework of Runge-Kutta methods

We interpret a wide range of flavors of Spectral Deferred Corrections (SDC) as Runge-Kutta methods (RKM). Using Butcher series, we show that the considered class of SDC methods achieve at least order p after p iterations compared to the underlying RKM, independently of the error discretisation chosen and the choice of nodes. For all collocation RKM, we analyse the phenomenon of order jumps in SDC iterations, where the order is increased by two at each iteration. We prove that it can be obtained by using appropriate inconsistent, implicit, parallelisable error discretisations. We also investigate the stability properties of the new SDC methods which can in general reduce to that of explicit RKM, but it can be improved by suitable combinations of error discretisations. We confirm the convergence analysis with numerical experiments and we apply relaxation RKM to derive SDC variants that conserve quadratic invariants.

math.NA↗

arXiv2025

High order integration of stochastic dynamics on Riemannian manifolds with frozen flow methods

We present a new class of numerical methods for solving stochastic differential equations with additive noise on general Riemannian manifolds with high weak order of accuracy. In opposition to the popular approach with projection methods, the proposed methods are intrinsic: they only rely on geometric operations and avoid coordinates and embeddings. We provide a robust and general convergence analysis and an algebraic formalism of exotic planar Butcher series for the computation of order conditions at any high order. To illustrate the methodology, an explicit method of second weak order is introduced, and several numerical experiments confirm the theoretical findings and extend the approach for the sampling of the invariant measure of Riemannian Langevin dynamics.

math.NA↗

arXiv2025

Efficient Langevin sampling with position-dependent diffusion

We introduce a numerical method for Brownian dynamics with position dependent diffusion tensor which is second order accurate for sampling the invariant measure while requiring only one force evaluation per timestep. Analysis of the sampling bias is performed using the algebraic framework of exotic aromatic Butcher-series. Numerical experiments confirm the theoretical order of convergence and illustrate the efficiency of the new method.

math.NA↗

arXiv2024

Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations

While backward error analysis does not generalise straightforwardly to the strong and weak approximation of stochastic differential equations, it extends for the sampling of ergodic dynamics. The calculation of the modified equation relies on tedious calculations and there is no expression of the modified vector field, in opposition to the deterministic setting. We uncover in this paper the Hopf algebra structures associated to the laws of composition and substitution of exotic aromatic S-series, relying on the new idea of clumping. We use these algebraic structures to provide the algebraic foundations of stochastic numerical analysis with S-series, as well as an explicit expression of the modified vector field as an exotic aromatic B-series.

math.NA↗

arXiv2023

Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling

B-series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-series and S-series. Precisely, we prove the relationship between the Grossman-Larson algebras over exotic and grafted forests and the corresponding duals to the Connes-Kreimer coalgebras and use it to study the natural composition laws on exotic S-series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge-Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

math.NA↗