Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0210320

A generalized Numerov method for linear second-order differential equations involving a first derivative term

V. I. Tselyaev
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Numerov method for linear second-order differential equations is generalized to include equations containing a first derivative term. The method presented has the same degree of accuracy as the ordinary Numerov sixth-order method. A general scheme of the application to the numerical solution of the Hartree-Fock equations is considered.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

V. I. Tselyaev. 2002-10-21. A generalized Numerov method for linear second-order differential equations involving a first derivative term. https://arxiv.org/abs/math/0210320

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

KEEP EXPLORING

Related papers

arXiv2026

PDEformer-2: A Versatile Foundation Model for Two-Dimensional Partial Differential Equations

Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with adequate accuracy, and limitations of the traditional solvers and specialized neural operators motivate the development of foundation models for solving PDEs. This paper introduces PDEformer-2, a versatile foundation model for two-dimensional PDEs. Based on our previous one-dimensional PDEformer-1 model, PDEformer-2 receives the PDE form as network input via computational graph representation, which has the flexibility to encode most common PDEs. The mesh-free predicted solutions can be directly queried at arbitrary spatio-temporal coordinates. A large (40TB) diverse dataset is employed to pretrain the current model, making it capable of simultaneously addressing PDEs with different symbolic forms, domain shapes, boundary conditions, number of variables, and time-dependency. Accurate zero-shot prediction is allowed for PDEs that resemble the pretraining ones. When adapted to new unseen PDEs, PDEformer-2 demonstrates faster learning than many specialized models, and has smaller errors given limited (less than 100) samples. Additionally, PDEformer-2 can be employed in the inverse problems thanks to its fast and differentiable nature and produces reasonable results in our experiments to recover coefficient scalars and fields of a PDE.

math.NA↗
arXiv2026

A divergence-free parametric finite element method for 3D Stokes equations on curved domains

The Stokes equations play an important role in the incompressible flow simulation. In this paper, a novel divergence-free parametric mixed finite element method is proposed for solving three-dimensional Stokes equations on domains with piecewise smooth boundaries. The flow velocity and pressure are discretized with high-order parametric Brezzi-Douglas-Marini elements and volume elements, respectively, on curved tetrahedral meshes. Utilizing the interior-penalty discontinuous Galerkin (IPDG) technique, we prove the inf-sup condition for the mixed finite element pair, and high-order optimal error estimates in the energy norm, with the help of the extension and transformation of the true solution to computational domain. Moreover, the discrete velocity is exactly divergence-free, meaning that $\Div\Bu_h=0$ holds in the curved computational domain. Numerical experiments are conducted to support the theoretical analyses.

math.NA↗
arXiv2026

A Reynolds- and Hartmann-semirobust hybrid method for magnetohydrodynamics

We propose and analyze a new method for the unsteady incompressible magnetohydrodynamics equations on convex domains with hybrid approximations of both vector-valued and scalar-valued fields. The proposed method is convection-semirobust, meaning that, for sufficiently smooth solutions, one can derive a priori estimates for the velocity and the magnetic field that do not depend on the inverse of the diffusion coefficients. This is achieved while at the same time providing relevant additional features, namely an improved order of convergence for the (asymptotic) diffusion-dominated regime, a small stencil (owing to the absence of inter-element penalty terms), and the possibility to significantly reduce the size of the algebraic problems through static condensation. The theoretical results are confirmed by a complete panel of numerical experiments.

math.NA↗