Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2402.13788

A Unifying Theory for Runge-Kutta-like Time Integrators: Convergence and Stability

Thomas Izgin
View original source ↗Read source document ↗Export metadata ↓

Abstract

The work deals with two major topics concerning the numerical analysis of Runge-Kutta-like (RK-like) methods, namely their stability and order of convergence. RK-like methods differ from additive RK methods in that their coefficients are allowed to depend on the solution and the step size. As a result of this, we also refer to them as non-standard additive RK (NSARK) methods. The first major part of this thesis is dedicated to providing a tool for deriving order conditions for NSARK methods. The proposed approach may yield implicit order conditions, which can be rewritten in explicit form using the NB-series of the stages. The obtained explicit order conditions can be further reduced using Gröbner bases computations. With the presented approach, it was possible for the first time to obtain conditions for the construction of 3rd and 4th order GeCo as well as 4th order MPRK schemes. Moreover, a new fourth order MPRK method is constructed using our theory and the order of convergence is validated numerically. The second major part is concerned with the stability of nonlinear time integrators preserving at least one linear invariant. We discuss how the given approach generalizes the notion of A-stability. We can prove that investigating the Jacobian of the generating map is sufficient to understand the stability of the nonlinear method in a neighborhood of the steady state. This approach allows for the first time the investigation of several modified Patankar. In the case of MPRK schemes, we compute a general stability function in a way that can be easily adapted to the case of PDRS. Finally, the approach from the theory of dynamical systems is used to derive a necessary condition for avoiding unrealistic oscillations of the numerical approximation.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Thomas Izgin. 2025-06-25. A Unifying Theory for Runge-Kutta-like Time Integrators: Convergence and Stability. https://doi.org/10.17170/kobra-202402059522

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators

In this work, we study the Kantorovich variant of max-min neural network operators, in which the operator kernel is defined in terms of sigmoidal functions. Our main aim is to demonstrate the $L^{p}$-convergence of these nonlinear operators for $1\leq p<\infty$, which makes it possible to obtain approximation results for functions that are not necessarily continuous. In addition, we will derive quantitative estimates for the rate of approximation in the $L^{p}$-norm. We will provide some explicit examples, studying the approximation of discontinuous functions with the max-min operator, and varying additionally the underlying sigmoidal function of the kernel. Further, we numerically compare the $L^{p}$-approximation error with the respective error of the Kantorovich variants of other popular neural network operators. As a final application, we show that the Kantorovich variant has advantages compared to the sampling variant of the max-min operator and Kantorovich variant of the max-product operator when it comes to approximate noisy functions as for instance biomedical ECG signals.

math.NA↗
arXiv2026

Quotient geometry of tensor ring decomposition

Differential geometries derived from tensor decompositions have been extensively studied and provided the foundations for a variety of efficient numerical methods. Despite the practical success of the tensor ring (TR) decomposition, its intrinsic geometry remains less understood, primarily due to the underlying ring structure and the resulting nontrivial gauge invariance. We establish the quotient geometry and immersed-submanifold structure of TR decomposition by imposing full-rank conditions on all unfolding matrices of the core tensors and capturing the gauge invariance. The intrinsic ring structure of TR leads to an analysis that is substantially different from other tensor formats. Additionally, for the uniform TR decomposition, where all core tensors are identical and the manifold structure is known, we derive explicit parameterizations for the vertical and horizontal spaces, which enable Riemannian optimization. Numerical experiments validate the developed geometries via tensor ring completion tasks.

math.NA↗
arXiv2026

Boundary elements for clamped Kirchhoff--Love plates

We present a Galerkin boundary element method for clamped Kirchhoff--Love plates with piecewise smooth boundary. It is a direct method based on the representation formula and requires the inversion of the single-layer operator, an application of the double-layer operator to the Dirichlet data, and, in the presence of a vertical load, an application of the Dirichlet trace of the Newton potential to that load. We present trace approximation spaces of arbitrary order, required for both the Dirichlet data and the unknown Neumann trace. Our boundary element method is quasi-optimal with respect to the natural trace norm and achieves optimal convergence order under minimal regularity assumptions. We provide explicit representations of all three integral operators and discuss the implementation of the appearing integrals. Numerical experiments for smooth and non-smooth domains confirm predicted convergence rates.

math.NA↗