Numerical Methods for Differential Equations and Applications

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

Deadline for manuscript submissions: 30 September 2024 | Viewed by 738

Special Issue Editor


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Guest Editor
Dyrecta Lab, Research Institute, Via Vescovo Simplicio 45, 70014 Conversano, Italy
Interests: numerical methods for differential equation; high order finite difference schemes; boundary value problems; singularly perturbed; singular; with discontinuous source; multipoint; Sturm–Liouville problems; multiparameter spectral problems; image processing; reaction and diffusion problems; colon cancer model; data mining

Special Issue Information

Dear Colleagues,

Differential equations with boundary conditions arise from many applications in engineering, biology, electrodynamics, quantum mechanics, physics. The numerical methods are essential for solving this kind of problems, especially, when the solutions are particularly stiff and high accuracy is difficult to be reached. Moreover, in many cases the problems can be singularly perturbed, singular, with discontinuous source terms, non-local, so that numerical methods can require to adopt a mesh adaptation to get an accurate solution. This special issue has the aim to collect latest development on the numerical methods solving boundary value problems that could also arise from real applications.

Dr. Giuseppina Settanni
Guest Editor

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Keywords

  • boundary value problems
  • ordinary differential equations
  • partial differential equations
  • non-local problems
  • singular problems
  • singularly perturbed problems
  • discontinuous source term
  • nonlinear differential equations
  • numerical approaches and applications

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Published Papers (1 paper)

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Research

23 pages, 1385 KiB  
Article
Strong Stability Preserving Two-Derivative Two-Step Runge-Kutta Methods
by Xueyu Qin, Zhenhua Jiang and Chao Yan
Mathematics 2024, 12(16), 2465; https://doi.org/10.3390/math12162465 - 9 Aug 2024
Viewed by 447
Abstract
In this study, we introduce the explicit strong stability preserving (SSP) two-derivative two-step Runge-Kutta (TDTSRK) methods. We propose the order conditions using Albrecht’s approach, comparing to the order conditions expressed in terms of rooted trees, these conditions present a more straightforward form with [...] Read more.
In this study, we introduce the explicit strong stability preserving (SSP) two-derivative two-step Runge-Kutta (TDTSRK) methods. We propose the order conditions using Albrecht’s approach, comparing to the order conditions expressed in terms of rooted trees, these conditions present a more straightforward form with fewer equations. Furthermore, we develop the SSP theory for the TDTSRK methods under certain assumptions and identify its optimal parameters. We also conduct a comparative analysis of the SSP coefficient among TDTSRK methods, two-derivative Runge-Kutta (TDRK) methods, and Runge-Kutta (RK) methods, both theoretically and numerically. The comparison reveals that the TDTSRK methods in the same order of accuracy have the most effective SSP coefficient. Numerical results demonstrate that the TDTSRK methods are highly efficient in solving the partial differential equation, and the TDTSRK methods can achieve the expected order of accuracy. Full article
(This article belongs to the Special Issue Numerical Methods for Differential Equations and Applications)
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