Numerical Methods and Modeling for Multiscale Problems and Applications

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

Deadline for manuscript submissions: 30 June 2025 | Viewed by 150

Special Issue Editors


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Guest Editor
Department of Mathematics and Computer Science, Università degli Studi di Catania, Catania, Italy
Interests: multiscale numerical methods; high-order schemes; implicit and semi-implicit schemes; order-adaptive procedures

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Università degli Studi di Catania, Catania, Italy
Interests: numerical methods for conservation laws; numerical methods for stiff problems; numerical methods for hyperbolic systems with a relaxation term; numerical methods for kinetic problems

Special Issue Information

Dear Colleagues,

The accurate and efficient modeling of multiscale systems remains one of the most compelling challenges in applied mathematics and computational science. From fluid dynamics and solid mechanics to environmental modeling and biological systems, multiscale phenomena are pervasive, characterized by interactions across spatial and temporal scales that defy traditional modeling and computational approaches.

This Special Issue aims to bring together innovative contributions on numerical methods and computational models that address the complexities inherent in multiscale problems. We welcome research that bridges the gap between theoretical developments and practical applications, fostering a dialogue between applied mathematicians, computational scientists, and engineers. Topics of interest include, but are not limited to, the following:

  • High-order numerical schemes for hyperbolic and dispersive partial differential equations (PDEs).      
  • Novel methods for coupled systems and hybrid multiscale models.
  • Computational strategies for systems exhibiting strong scale separation.
  • Applications of multiscale modeling in fluid dynamics, continuum mechanics, and environmental systems. 
  • Advances in numerical solvers for non-linear PDEs in multiscale frameworks.

The scope of this Special Issue is deliberately broad to encompass a wide range of approaches and applications, with a particular focus on innovative methodologies that push the boundaries of current computational capabilities. Contributions highlighting real-world applications, especially in the natural sciences and engineering, are strongly encouraged.

We invite original research papers, comprehensive reviews, and methodological studies that explore the forefront of multiscale modeling and numerical analysis.

Dr. Emanuele Macca
Dr. Sebastiano Boscarino
Guest Editors

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Keywords

  • multiscale numerical methods
  • high-order schemes
  • coupled multiscale systems
  • implicit and semi-implicit approaches
  • computational applications in multiscale phenomena

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Published Papers (1 paper)

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Research

31 pages, 1363 KiB  
Article
A Fractional PDE-Based Model for Nerve Impulse Transport Solved Using a Conforming Virtual Element Method: Application to Prosthetic Implants
by Zaffar Mehdi Dar, Chandru Muthusamy and Higinio Ramos
Axioms 2025, 14(6), 398; https://doi.org/10.3390/axioms14060398 - 23 May 2025
Abstract
The main objective of this study is to present a fundamental mathematical model for nerve impulse transport, based on the underlying physical phenomena, with a straightforward application in describing the functionality of prosthetic devices. The governing equation of the resultant model is a [...] Read more.
The main objective of this study is to present a fundamental mathematical model for nerve impulse transport, based on the underlying physical phenomena, with a straightforward application in describing the functionality of prosthetic devices. The governing equation of the resultant model is a two-dimensional nonlinear partial differential equation with a time-fractional derivative of order α ∈ (0,1). A novel and effective numerical approach for solving this fractional-order problem is constructed based on the virtual element method. Three basic technical building blocks form the basis of our methodology: the regularity theory related to nonlinearity, discrete maximal regularity, and a fractional variant of the Grünwald--Letnikov approximation. By utilizing these components, along with the energy projection operator, a fully discrete virtual element scheme is formulated in such a way that it ensures stability and consistency. We establish the uniqueness and existence of the approximate solution. Numerical findings confirm the convergence in the L2-norm and H1-norm on both uniform square and regular Voronoi meshes, confirming the effectiveness of the proposed model and method, and their potential to support the efficient design of sensory prosthetics. Full article
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