Special Issue "Geometric Numerical Integration"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 July 2019

Special Issue Editor

Guest Editor
Prof. Dr. Jan L. Cieśliński

University of Białystok, Faculty of Physics, Białystok, Poland
Website | E-Mail
Interests: integrable systems of differential and difference equations; differential geometry; geometric numerical integration; numerical approximation of elementary functions

Special Issue Information

Dear Colleagues,

The construction of numerical solutions to differential equations is one of the most important activities in applied mathematics. In the last 30 years, the focus in this field shifted from general-purpose algorithms to special-purpose methods tailored to preserve special features of given classes of equations. Looking from the present perspective one can notice many predecessors even in earlier years. In astronomy and molecular dynamics numerical schemes preserving the total energy and angular momentum were always quite popular. The famous leap-frog scheme turned out to be symplectic what is one of the reasons for its good performance. The preservation of structural properties by the numerical discretization is of great advantage for rendering good qualitative features of solution trajectories, also in the asymptotic region.

The purpose of this Special Issue is to report and review recent developments concerning various approaches to structure-preserving integration of differential equations. First of all, we solicit papers on classical topics of geometric numerical integration of ordinary differential equations, including symplectic integrators and numerical methods preserving first integrals, volume, symmetries, time-reversibility or other characteristics. Nonstandard finite difference schemes preserving qualitative properties of differential equations and geometric numerical methods for partial differential equations, known as compatible discretizations or mimetic methods, fall also within the scope of this issue. Contributions on related topics, like time scales approach or integrable discretizations, are welcome as well, provided that they focus on numerical aspects.

All submitted papers will be peer-reviewed and selected on the basis of both their quality and their relevance to the theme of this Special Issue.

Prof. Dr. Jan L. Cieśliński
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 850 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • structure preserving numerical schemes
  • symplectic integrators for Hamiltonian systems
  • variational integrators
  • numerical methods preserving first integrals
  • discrete gradient schemes
  • exponential integrators
  • numerical methods on manifolds, including Lie groups
  • numerical methods for highly oscillatory systems
  • nonstandard numerical schemes
  • integrable discretizations of integrable systems
  • compatible spatial discretizations
  • mimetic finite difference methods for partial differential equations

Published Papers (1 paper)

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Review

Open AccessReview
Line Integral Solution of Hamiltonian PDEs
Mathematics 2019, 7(3), 275; https://doi.org/10.3390/math7030275
Received: 8 January 2019 / Revised: 8 March 2019 / Accepted: 12 March 2019 / Published: 18 March 2019
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Abstract
In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger [...] Read more.
In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries equation, to illustrate the main features of this novel approach. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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