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Axioms
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The Numerical Analysis and Its Application, 2nd Edition
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The Numerical Analysis and Its Application, 2nd Edition

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 30 June 2026 | Viewed by 1493

Special Issue Editor

Dr. Janis Bajars

E-Mail Website
Guest Editor
Faculty of Physics, Mathematics and Optometry, University of Latvia, Riga, Latvia
Interests: numerical methods for differential and integral equations; dynamical systems; crystal lattice models; wave phenomena; data-driven algorithms
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The value of applied numerical analysis and the power of computational modelling of real-life phenomena cannot be understated. Their applications range from computer, physical, life, and social sciences to engineering and emerging technologies. Further research in advancing applied and computational mathematics is indispensable for resolving real-life, engineering, technological, and scientific problems.

This Special Issue is dedicated to illustrating the numerical analysis and development of novel computational techniques as well as algorithms advancing diverse applications of mathematical modelling, theoretical physics, dynamical systems, differential and integral equations, optimization methods, structure-preserving methods, learning algorithms, and data-driven methods, but is not limited to these research fields. With this Special Issue, we aim to either appraise the improvements of existing computational methods or demonstrate new algorithmic and computational methodologies, including learning and data-driven methods. For this Special Issue, theoretical and applied contributions with extensive numerical studies for different applications are welcome. Contributions demonstrating the improved analytical results and computational efficiency of already-existing or new algorithms, as well as the discovery of new phenomena driven by the development of numerical algorithms, are of particular interest.

Dr. Janis Bajars
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 250 words) can be sent to the Editorial Office for assessment.

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. Axioms 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 2400 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

  • applied numerical analysis
  • computational mathematics
  • numerical methods for differential and integral equations
  • geometric numerical integration
  • data-driven dynamical systems
  • physics-informed learning
  • optimization techniques in mathematical modelling

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Related Special Issue

Published Papers (2 papers)

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Research

22 pages, 1232 KB  
Article
An Energy-Stable S-SAV Finite Element Method for the Generalized Poisson-Nernst-Planck Equation
by Maoqin Yuan, Junde Liu, Peng Ma and Mingyang Li
Axioms 2026, 15(2), 126; https://doi.org/10.3390/axioms15020126 - 7 Feb 2026
Viewed by 530
Abstract
Designing structure-preserving numerical schemes for the generalized Poisson-Nernst-Planck (PNP) system is challenging due to its inherent strong nonlinearity and coupling. In this paper, we propose a class of efficient, unconditional energy-stable schemes based on the Stabilized Scalar Auxiliary Variable (S-SAV) framework combined with [...] Read more.
Designing structure-preserving numerical schemes for the generalized Poisson-Nernst-Planck (PNP) system is challenging due to its inherent strong nonlinearity and coupling. In this paper, we propose a class of efficient, unconditional energy-stable schemes based on the Stabilized Scalar Auxiliary Variable (S-SAV) framework combined with the finite element method. We construct both first-order (BE-S-SAV) and second-order (BDF2-S-SAV) fully discrete schemes. A distinguishing feature of our approach is the use of a linear decomposition strategy, which decouples the complex nonlinear system into a sequence of linear, constant-coefficient elliptic equations at each time step. This significantly reduces computational complexity by avoiding expensive nonlinear iterations. We provide rigorous theoretical proofs demonstrating that the proposed schemes are unconditionally energy stable and strictly preserve mass conservation. Numerical experiments satisfy the theoretical analysis, confirming optimal convergence rates and demonstrating robust preservation of mass conservation and modified energy stability in the tested regimes. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application, 2nd Edition)
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38 pages, 542 KB  
Article
The Extended Second APG Method for Constrained DC Problems
by Ziye Liu, Huitao Ke and Chunguang Liu
Axioms 2026, 15(1), 7; https://doi.org/10.3390/axioms15010007 - 24 Dec 2025
Viewed by 492
Abstract
In this paper, we develop the extended proximal gradient algorithm with Nesterov’s second acceleration (EAPGs) for constrained difference-of-convex (DC) optimization problems. EAPGs has two key links to existing methods: it extends APGs (for unconstrained DC problems) by adopting [...] Read more.
In this paper, we develop the extended proximal gradient algorithm with Nesterov’s second acceleration (EAPGs) for constrained difference-of-convex (DC) optimization problems. EAPGs has two key links to existing methods: it extends APGs (for unconstrained DC problems) by adopting the constraint handling idea from Auslender’s ESQM, and serves as a variant of ESQMe with extrapolation replaced by Nesterov’s second acceleration. Under basic assumptions, we establish the subsequential convergence of EAPGs. By introducing a restart technique and leveraging the Kurdyka–Łojasiewicz (KL) property of a suitable potential function, we further prove its global convergence, analyze its convergence rate, and do so under weaker conditions than those for APGs. Additionally, we propose EAPGsr by adding practical restart criteria to EAPGs. Numerical experiments verify the criteria’s efficiency and show that EAPGsr performs well against state-of-the-art methods for constrained and unconstrained DC problems. Full article
(This article belongs to the Special Issue The Numerical Analysis and Its Application, 2nd Edition)
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