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Symmetry
Special Issues
Mesh Methods - Numerical Analysis and Experiments
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Special Issue "Mesh Methods - Numerical Analysis and Experiments"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 31 December 2019.

Special Issue Editors

Prof. Viktor A. Rukavishnikov
E-Mail Website
Guest Editor
Computing Center of Far-Eastern Branch, Russian Academy of Sciences, Kim-Yu-Chen Str. 65, Khabarovsk 680000, Russia
Interests: boundary value problems with singularity, numerical methods in electrodynamics, hydrodynamics and theory of elasticity
Prof. Pedro M. Lima
E-Mail Website
Guest Editor
Instituto Superior Tecnico, University of Lisbon, Lisbon, Portugal
Interests: Singular boundary value problems; integro-differential equations; high-order methods
Prof. Ildar B. Badriev
E-Mail Website
Guest Editor
Kazan Federal University, 18 Kremlyovskaya Street, Kazan 420008, Russian Federation
Interests: Numerical solution of the inetial boundary value problems; difference method; mathematical simulation

Special Issue Information

Dear Colleagues,

Mathematical models of different natural processes are described by differential equations, systems of PDEs and integral equations. In most cases, the exact solution of such problems cannot be determined, so we have to use mesh methods to calculate an approximate solution using high performance computational complexes. These methods include the finite element method, the finite difference method, the finite volume method and combined methods.

In this Special Issue, we propose to publish qualitative works on theoretical studies of grid methods on approximation, stability, and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. New methods for boundary value problems with singularity, with a complex geometry of the domain boundary, and for non-linear equations are of particular interest. Articles concerning analysis of the numerical methods developed for the computation of mathematical models in different areas of applied science and engineering applications will be welcome.

As a rule, symmetry ideas are present in the computational schemes and make the process harmonious and effective.

Prof. Viktor A. Rukavishnikov
Prof. Pedro M. Lima
Prof. Ildar B. Badriev
Guest Editors

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. Symmetry 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 1400 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

  • finite element method
  • difference method
  • finite volume method
  • numerical experiments
  • numerical analysis
  • singularity
  • symmetry

Published Papers (2 papers)

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Research

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Open AccessArticle
Numerical Method for Dirichlet Problem with Degeneration of the Solution on the Entire Boundary
by Viktor A. Rukavishnikov and Elena I. Rukavishnikova
Symmetry 2019, 11(12), 1455; https://doi.org/10.3390/sym11121455 - 26 Nov 2019
Abstract
The finite element method (FEM) with a special graded mesh is constructed for the Dirichlet boundary value problem with degeneration of the solution on the entire boundary of the two-dimensional domain. A comparative numerical analysis is performed for the proposed method and the [...] Read more.
The finite element method (FEM) with a special graded mesh is constructed for the Dirichlet boundary value problem with degeneration of the solution on the entire boundary of the two-dimensional domain. A comparative numerical analysis is performed for the proposed method and the classical finite element method for a set of model problems in symmetric domain. Experimental confirmation of theoretical estimates of accuracy is obtained and conclusions are made. Full article
(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)
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Open AccessArticle
Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes
by Nikhil Anand , Neda Ebrahimi Pour , Harald Klimach and Sabine Roller
Symmetry 2019, 11(9), 1126; https://doi.org/10.3390/sym11091126 - 05 Sep 2019
Abstract
We investigate the suitability of the Brinkman penalization method in the context of a high-order discontinuous Galerkin scheme to represent wall boundaries in compressible flow simulations. To evaluate the accuracy of the wall model in the numerical scheme, we use setups with symmetric [...] Read more.
We investigate the suitability of the Brinkman penalization method in the context of a high-order discontinuous Galerkin scheme to represent wall boundaries in compressible flow simulations. To evaluate the accuracy of the wall model in the numerical scheme, we use setups with symmetric reflections at the wall. High-order approximations are attractive as they require few degrees of freedom to represent smooth solutions. Low memory requirements are an essential property on modern computing systems with limited memory bandwidth and capability. The high-order discretization is especially useful to represent long traveling waves, due to their small dissipation and dispersion errors. An application where this is important is the direct simulation of aeroacoustic phenomena arising from the fluid motion around obstacles. A significant problem for high-order methods is the proper definition of wall boundary conditions. The description of surfaces needs to match the discretization scheme. One option to achieve a high-order boundary description is to deform elements at the boundary into curved elements. However, creating such curved elements is delicate and prone to numerical instabilities. Immersed boundaries offer an alternative that does not require a modification of the mesh. The Brinkman penalization is such a scheme that allows us to maintain cubical elements and thereby the utilization of efficient numerical algorithms exploiting symmetry properties of the multi-dimensional basis functions. We explain the Brinkman penalization method and its application in our open-source implementation of the discontinuous Galerkin scheme, Ateles. The core of this presentation is the investigation of various penalization parameters. While we investigate the fundamental properties with one-dimensional setups, a two-dimensional reflection of an acoustic pulse at a cylinder shows how the presented method can accurately represent curved walls and maintains the symmetry of the resulting wave patterns. Full article
(This article belongs to the Special Issue Mesh Methods - Numerical Analysis and Experiments)
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

MODIFIED MULTIGRID METHOD FOR THE NAVIER-STOKES EQUATIONS WITH THE HSS-SMOOTHERS
E. Andreeva, V. Bavin, T. Martynova, G. Muratova

A modification of the multigrid method (MGM) is considered for discretized systems of partial differential equations (PDEs) which arise from finite difference approximation of the incompressible Navier-Stokes equations. After discretization and linearization of the equations, systems of linear algebraic equations (SLAE) with a strongly non-Hermitian matrix is obtained. Hermitian and Skew-Hermitian Splitting (HSS) smoothers are used in the MGM for solving of the SLAE. Parallel algebraic multigrid method with GPGPU technology is used. Some numerical results are presented.

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