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Symmetry
Special Issues
Recent Progress in Studies of Stability of Numerical Schemes
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Recent Progress in Studies of Stability of Numerical Schemes

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 12244

Special Issue Editors

Prof. Dr. Taras I. Lakoba

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Guest Editor
Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA
Interests: stability of numerical methods for nonlinear waves
Dr. Sanda Micula

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Kogalniceanu St., 400084 Cluj Napoca, Romania
Interests: numerical methods; integral equations; approximation theory
Dr. Marcel-Adrian Serban

E-Mail Website
Guest Editor
Department of Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
Interests: operator theory; ordinary differential equations; integral equations

Special Issue Information

Dear Colleagues,

The stability of numerical methods for solving differential equations is a fundamental property that is necessary for the method to produce a valid solution. There are various concepts for numerical stability in the case of ordinary differential equations or partial differential equations.

For ODEs, A-stability can be considered, which is related to some concept of stability in the dynamical systems sense. Continuous dependence of data studies, Grönwall-type inequalities, in differential or integral form, can also be employed to establish stability results.

The von Neumann analysis is the standard tool for establishing the stability of a numerical scheme for PDEs. However, its limitation is that it strictly applies only to linear equations with spatially constant coefficients and periodic boundary conditions. Extensions of the von Neumann stability analysis to other, i.e. nonlinear or non-constant, coefficient equations usually appeal to the principle of frozen coefficients, whereby near a given spatial location, the coefficients of the equation are approximately considered as being constant. Such an approach is able to detect instability if it originates locally within the numerical grid. However, it is less successful in detecting instabilities that originate either due to non-periodic boundary conditions or due to some nontrivial scattering of Fourier harmonics on the non-constant background. In the past decade or so, a number of studies have addressed the stability of numerical methods (e.g., the split-step method for dispersive/parabolic equations, or the method of characteristics for hyperbolic equations) with approaches that go beyond the von Neumann analysis.

The purpose of this Special Issue is to collect new theoretical and numerical studies concerning the techniques applied for proving the stability or instability of numerical schemes, which extend or improve the known results. Contributions highlighting structure-preserving and symmetry-preserving numerical methods are particularly welcome.


Prof. Dr. Taras I. Lakoba
Dr. Sanda Micula
Dr. Marcel-Adrian Serban
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 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. 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 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

  • stability of numerical methods
  • numerical stability
  • method of characteristics
  • interpolation of curved characteristics
  • non-periodic boundary conditions
  • symmetry- and invariant-preserving schemes

Published Papers (7 papers)

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Editorial

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3 pages, 681 KiB  
Editorial
Recent Progress in Studies of Stability of Numerical Schemes
by Taras I. Lakoba and Sanda Micula
Symmetry 2022, 14(12), 2692; https://doi.org/10.3390/sym14122692 - 19 Dec 2022
Viewed by 1001
Abstract
Applications and modeling of various phenomena in all areas of scientific research require finding numerical solutions for differential, partial differential, integral, or integro-differential equations. In addition to proving theoretical convergence and giving error estimates, stability of numerical methods for such operator equations is [...] Read more.
Applications and modeling of various phenomena in all areas of scientific research require finding numerical solutions for differential, partial differential, integral, or integro-differential equations. In addition to proving theoretical convergence and giving error estimates, stability of numerical methods for such operator equations is a fundamental property that it is necessary for the method to produce a valid solution. This Special Issue focuses on new theoretical and numerical studies concerning the techniques used for proving stability or instability of numerical schemes, which extend or improve known results. It also includes applications to non-linear physical, chemical, and engineering systems, arising in dynamics of waves, diffusion, or transport problems. Full article
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)

Research

Jump to: Editorial

10 pages, 265 KiB  
Article
Ulam–Hyers Stability via Fixed Point Results for Special Contractions in b-Metric Spaces
by Monica-Felicia Bota and Sanda Micula
Symmetry 2022, 14(11), 2461; https://doi.org/10.3390/sym14112461 - 20 Nov 2022
Cited by 7 | Viewed by 994
Abstract
In this paper, we present some fixed point results for Subrahmanyan contraction in the setting of a b-metric space. We consider the case of multivalued operators. We also deduce the Ulam–Hyers stability property of the fixed point inclusion. The notion of b [...] Read more.
In this paper, we present some fixed point results for Subrahmanyan contraction in the setting of a b-metric space. We consider the case of multivalued operators. We also deduce the Ulam–Hyers stability property of the fixed point inclusion. The notion of b-metric generalizes the one of a metric, as in the third condition, the right-hand side is multiplied by a real number greater than 1. We remark that the second axiom, i.e., the one which shows the symmetry of the b-metric, remains unchanged. The findings presented in this paper extend some recent results which were proved in the context of a metric space. Some open questions are presented at the end of the paper. Full article
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)
15 pages, 1100 KiB  
Article
Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods
by Leila Torkzadeh, Hassan Ranjbar, Sanda Micula and Kazem Nouri
Symmetry 2022, 14(11), 2413; https://doi.org/10.3390/sym14112413 - 15 Nov 2022
Cited by 1 | Viewed by 1181
Abstract
We introduce two approaches by modifying split-step exponential schemes to study stochastic differential equations. Under the Lipschitz condition and linear-growth bounds, it is shown that our explicit schemes converge to the solution of the corresponding stochastic differential equations with the order 1.0 in [...] Read more.
We introduce two approaches by modifying split-step exponential schemes to study stochastic differential equations. Under the Lipschitz condition and linear-growth bounds, it is shown that our explicit schemes converge to the solution of the corresponding stochastic differential equations with the order 1.0 in the mean-square sense. The mean-square stability of our methods is investigated through some linear stochastic test systems. Additionally, asymptotic mean-square stability is analyzed for the two-dimensional system with symmetric and asymmetric coefficients and driven by two commutative noise terms. In particular, we prove that our methods are mean-square stable for any step-size. Finally, some numerical experiments are carried out to confirm the theoretical results. Full article
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)
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18 pages, 3724 KiB  
Article
Stable Difference Schemes with Interpolation for Delayed One-Dimensional Transport Equation
by Karthick Sampath, Subburayan Veerasamy and Ravi P. Agarwal
Symmetry 2022, 14(5), 1046; https://doi.org/10.3390/sym14051046 - 19 May 2022
Cited by 5 | Viewed by 1363
Abstract
In this article, we consider the one-dimensional transport equation with delay and advanced arguments. A maximum principle is proven for the problem considered. As an application of the maximum principle, the stability of the solution is established. It is also proven that the [...] Read more.
In this article, we consider the one-dimensional transport equation with delay and advanced arguments. A maximum principle is proven for the problem considered. As an application of the maximum principle, the stability of the solution is established. It is also proven that the solution’s discontinuity propagates. Finite difference methods with linear interpolation that are conditionally stable and unconditionally stable are presented. This paper presents applications of unconditionally stable numerical methods to symmetric delay arguments and differential equations with variable delays. As a consequence, the matrices of the difference schemes are asymmetric. An illustration of the unconditional stable method is provided with numerical examples. Solution graphs are drawn for all the problems. Full article
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)
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14 pages, 4925 KiB  
Article
Solution of the 1D KPZ Equation by Explicit Methods
by Okhunjon Sayfidinov, Gabriella Bognár and Endre Kovács
Symmetry 2022, 14(4), 699; https://doi.org/10.3390/sym14040699 - 29 Mar 2022
Cited by 6 | Viewed by 2006
Abstract
The Kardar–Parisi-Zhang (KPZ) equation is examined using the recently published leapfrog–hopscotch (LH) method as well as the most standard forward time centered space (FTCS) scheme and the Heun method. The methods are verified by reproducing an analytical solution. The performance of each method [...] Read more.
The Kardar–Parisi-Zhang (KPZ) equation is examined using the recently published leapfrog–hopscotch (LH) method as well as the most standard forward time centered space (FTCS) scheme and the Heun method. The methods are verified by reproducing an analytical solution. The performance of each method is then compared by calculating the average and the maximum differences among the results and displaying the runtimes. Numerical tests show that due to the special symmetry in the time–space discretisation, the new LH method clearly outperforms the other two methods. In addition, we discuss the effect of different parameters on the solutions. Full article
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)
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21 pages, 1663 KiB  
Article
Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion
by Siwei Duo, Taras I. Lakoba and Yanzhi Zhang
Symmetry 2021, 13(8), 1394; https://doi.org/10.3390/sym13081394 - 01 Aug 2021
Cited by 2 | Viewed by 2164
Abstract
We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian (−Δ)α/2. The linear stability analysis [...] Read more.
We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian (−Δ)α/2. The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power α∈[1,2] but unstable for α∈(0,1). In the focusing case, they can be linearly unstable for any α∈(0,2]. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as α∈(1,2] decreases, the solution evolves towards an increasingly localized pulse existing on the background of a “sea” of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For α≤1, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for α<1, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities. Full article
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)
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9 pages, 1887 KiB  
Article
Numerical Stability Investigations of the Method of Fundamental Solutions Applied to Wave-Current Interactions Using Generating-Absorbing Boundary Conditions
by Mohammed Loukili, Denys Dutykh, Kamila Kotrasova and Dezhi Ning
Symmetry 2021, 13(7), 1153; https://doi.org/10.3390/sym13071153 - 27 Jun 2021
Cited by 11 | Viewed by 2096
Abstract
In this paper, the goal is to revolve around discussing the stability of the Method of Fundamental Solutions (MFS) for the use case of wave-current interactions. Further, the reliability of Generating-Absorbing Boundary Conditions (GABCs) applied to the wave-current interactions is investigated using the [...] Read more.
In this paper, the goal is to revolve around discussing the stability of the Method of Fundamental Solutions (MFS) for the use case of wave-current interactions. Further, the reliability of Generating-Absorbing Boundary Conditions (GABCs) applied to the wave-current interactions is investigated using the Method of Fundamental Solutions (MFS), in a Numerical Wave Tank (NWT) within the potential theory where the main regular manifestations are the periodicity, and symmetry of traveling waves. Besides, the investigations cover different aspects of currents (coplanar current, without current, and opposing current), and also different water depths. Furthermore, the accuracy and stability of the numerical method (MFS) used in this work is evaluated for different locations and numbers of source points. Full article
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)
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