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Mathematics
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Geometric Numerical Integration
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Geometric Numerical Integration

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2019) | Viewed by 23820

Special Issue Editor

Prof. Dr. Jan L. Cieśliński

E-Mail Website
Guest Editor
Faculty of Physics, University of Białystok, 15-328 Białystok, Poland
Interests: integrable systems; solitons; differential and difference equations; time scales; geometric numerical integration; differential and discrete geometry; Lie symmetries; Clifford algebras; 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 submissions that pass pre-check are 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 semimonthly 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 2600 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

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Published Papers (7 papers)

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Research

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27 pages, 629 KiB  
Article
Energetic-Property-Preserving Numerical Schemes for Coupled Natural Systems
by Mizuka Komatsu, Shunpei Terakawa and Takaharu Yaguchi
Mathematics 2020, 8(2), 249; https://doi.org/10.3390/math8020249 - 14 Feb 2020
Cited by 1 | Viewed by 2657
Abstract
In this paper, we propose a method for deriving energetic-property-preserving numerical schemes for coupled systems of two given natural systems. We consider the case where the two systems are interconnected by the action–reaction law. Although the derived schemes are based on the discrete [...] Read more.
In this paper, we propose a method for deriving energetic-property-preserving numerical schemes for coupled systems of two given natural systems. We consider the case where the two systems are interconnected by the action–reaction law. Although the derived schemes are based on the discrete gradient method, in the case under consideration, the equation of motion is not of the usual form represented by using the skew-symmetric matrix. Hence, the energetic-property-preserving schemes cannot be obtained by straightforwardly using the discrete gradient method. We show numerical results for two coupled systems as examples; the first system is a combination of the wave equation and the elastic equation, and the second is of the mass–spring system and the elastic equation. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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13 pages, 769 KiB  
Article
Locally Exact Integrators for the Duffing Equation
by Jan L. Cieśliński and Artur Kobus
Mathematics 2020, 8(2), 231; https://doi.org/10.3390/math8020231 - 10 Feb 2020
Cited by 5 | Viewed by 1985
Abstract
A numerical scheme is said to be locally exact if after linearization (around any point) it becomes exact. In this paper, we begin with a short review on exact and locally exact integrators for ordinary differential equations. Then, we extend our approach on [...] Read more.
A numerical scheme is said to be locally exact if after linearization (around any point) it becomes exact. In this paper, we begin with a short review on exact and locally exact integrators for ordinary differential equations. Then, we extend our approach on equations represented in the so called linear gradient form, including dissipative systems. Finally, we apply this approach to the Duffing equation with a linear damping and without external forcing. The locally exact modification of the discrete gradient scheme preserves the monotonicity of the Lyapunov function of the discretized equation and is shown to be very accurate. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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19 pages, 508 KiB  
Article
Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation
by Philipp Bader, Sergio Blanes and Fernando Casas
Mathematics 2019, 7(12), 1174; https://doi.org/10.3390/math7121174 - 3 Dec 2019
Cited by 30 | Viewed by 5647
Abstract
A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient [...] Read more.
A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Padé approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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31 pages, 1512 KiB  
Article
Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities
by Tomasz M. Tyranowski and Mathieu Desbrun
Mathematics 2019, 7(9), 861; https://doi.org/10.3390/math7090861 - 18 Sep 2019
Cited by 5 | Viewed by 3353
Abstract
In this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the [...] Read more.
In this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the “Hamiltonian” equations of motion can be formulated as an index-1 differential-algebraic system. We also construct variational Runge–Kutta methods and analyze their properties. The general properties of Runge–Kutta methods depend on the “velocity” part of the Lagrangian. If the “velocity” part is also linear in the position coordinate, then we show that non-partitioned variational Runge–Kutta methods are equivalent to integration of the corresponding first-order Euler–Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge–Kutta method are retained. If the “velocity” part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We verified our results through numerical experiments for various dynamical systems. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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52 pages, 2846 KiB  
Article
R-Adaptive Multisymplectic and Variational Integrators
by Tomasz M. Tyranowski and Mathieu Desbrun
Mathematics 2019, 7(7), 642; https://doi.org/10.3390/math7070642 - 18 Jul 2019
Cited by 6 | Viewed by 2890
Abstract
Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation but redistribute them over time to follow the areas [...] Read more.
Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this paper, we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations, and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. Numerical results for the Sine–Gordon equation are also presented. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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Review

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30 pages, 1271 KiB  
Review
Geometric Numerical Integration in Ecological Modelling
by Fasma Diele and Carmela Marangi
Mathematics 2020, 8(1), 25; https://doi.org/10.3390/math8010025 - 20 Dec 2019
Cited by 12 | Viewed by 2815
Abstract
A major neglected weakness of many ecological models is the numerical method used to solve the governing systems of differential equations. Indeed, the discrete dynamics described by numerical integrators can provide spurious solution of the corresponding continuous model. The approach represented by the [...] Read more.
A major neglected weakness of many ecological models is the numerical method used to solve the governing systems of differential equations. Indeed, the discrete dynamics described by numerical integrators can provide spurious solution of the corresponding continuous model. The approach represented by the geometric numerical integration, by preserving qualitative properties of the solution, leads to improved numerical behaviour expecially in the long-time integration. Positivity of the phase space, Poisson structure of the flows, conservation of invariants that characterize the continuous ecological models are some of the qualitative characteristics well reproduced by geometric numerical integrators. In this paper we review the benefits induced by the use of geometric numerical integrators for some ecological differential models. Full article
(This article belongs to the Special Issue Geometric Numerical Integration)
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28 pages, 1480 KiB  
Review
Line Integral Solution of Hamiltonian PDEs
by Luigi Brugnano, Gianluca Frasca-Caccia and Felice Iavernaro
Mathematics 2019, 7(3), 275; https://doi.org/10.3390/math7030275 - 18 Mar 2019
Cited by 13 | Viewed by 3225
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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