Análisis de estabilidad y convergencia en métodos numéricos de diferencias finitas para ecuaciones de calor y onda en una dimensión espacial

Martha Rebeca Cevallos Taimal; Mishell Dayana Usca Chicaiza; Josselyn Dayanna Caizaluisa Lara; Jessica Gabriela Grandes Padilla; Judith Johanna Portilla Vásquez

doi:10.70577/asce.v5i1.637

Authors

Martha Rebeca Cevallos Taimal Universidad Central del Ecuador; Pontificia Universidad Católica del Ecuador https://orcid.org/0009-0004-9211-4191
Mishell Dayana Usca Chicaiza Universidad Central del Ecuador https://orcid.org/0009-0004-2704-5628
Josselyn Dayanna Caizaluisa Lara Universidad Central del Ecuador https://orcid.org/0009-0001-6617-3735
Jessica Gabriela Grandes Padilla Universidad Central del Ecuador; ; Pontificia Universidad Católica del Ecuador https://orcid.org/0009-0004-0807-698X
Judith Johanna Portilla Vásquez Universidad de Guayaquil https://orcid.org/0009-0007-0182-1280

DOI:

https://doi.org/10.70577/asce.v5i1.637

Keywords:

Partial differential equations; Finite difference methods; Numerical stability; Convergence; Heat equation; Wave equation.

Abstract

Parabolic and hyperbolic partial differential equations are essential for modeling diffusion and propagation processes, such as heat transfer and wave dynamics. Therefore, numerical methods are used in real applications, among which finite differences stand out for their simplicity and efficiency in one-dimensional domains with uniform meshing and Dirichlet or Neumann boundaries. This work systematically analyzes the theoretical and numerical evidence published between 2020 and 2025 on stability, order of convergence, and computational cost of explicit, implicit, and semi-implicit schemes applied to the heat equation and, complementarily, to the 1D wave equation. The review was carried out through a structured analysis in renowned scientific databases such as SCOPUS, Web of Science, SciELO, and Google Scholar, where thirty open-access articles were selected through an analysis of stability, error estimates, and comparative tests in the numerical field. The results indicate that classical explicit schemes have restrictions due to CFL-type criteria, which affects overall efficiency when fine meshes are required. On the other hand, implicit and semi-implicit methods, with special emphasis on Crank–Nicolson, offer greater second-order convergence through the execution of algebraic processes and an increase in the cost per time step. Likewise, it is observed that high-order compact schemes and stabilized explicit variants can provide competitive trade-offs between accuracy and efficiency under standard conditions. The choice of scheme should be based on a comprehensive criterion that combines practical stability, accuracy, and computational cost according to the equation and discretization regime.

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