{
  "abstract": "We study three multistep implicit–explicit (IMEX) time discretizations coupled with a local discontinuous Galerkin (LDG) spatial discretization for a class of one-dimensional nonlinear fourth-order partial differential equations (PDEs), with particular emphasis on the Kuramoto–Sivashinsky (KS) equation. The nonlinear convection is treated explicitly, whereas the linear second-, third-, and fourth-derivative terms are advanced implicitly. Under the quasi-uniform mesh assumption and a uniform boundedness assumption on the exact and numerical solutions, energy estimates yield L2 stability for the three schemes under a time-step restriction τ≤τ0, where τ0 is independent of the spatial mesh size. For all three schemes, a coupled LDG elliptic projection and energy estimates give the optimal spatial rate O(hN+1) together with temporal rates O(τ), O(τ2), and O(τ3), provided the multistep starting values have the corresponding accuracy. The error analysis uses a coupled LDG elliptic projection so that the discrete auxiliary error equations remain homogeneous; the residual lower-order projection terms, including the β-term, are estimated explicitly. An independent periodic manufactured-solution test verifies temporal rates approaching one, two, and three. A separate periodic spatial-refinement test at T=0.5 with the third-order formula recovers the expected O(hN+1) spatial behavior. For a long-time periodic KS test, kinetic-energy histories and spatial spectra are compared under both time-step and mesh refinement. The latter diagnostics are interpreted as robustness checks over a finite observation window rather than as evidence of a statistically stationary chaotic state or pointwise long-time convergence.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Amjad G. Alharbi"
    },
    {
      "affs": [
        0
      ],
      "name": "Tarek Aboelenen",
      "orcid": "0000-0003-3332-7950"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aveqfd2maqx5v.bib",
  "cited_by": 0,
  "doi": "10.3390/math14213950",
  "fulltext_url": null,
  "id": "aveqfd2maqx5v",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/21/3950",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/aveqfd2maqx5v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14213950",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/21/3950"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Stability and Error Analysis of Multistep IMEX Local Discontinuous Galerkin Methods for One-Dimensional Nonlinear Fourth-Order Evolution Problems",
  "title_text": "Stability and Error Analysis of Multistep IMEX Local Discontinuous Galerkin Methods for One-Dimensional Nonlinear Fourth-Order Evolution Problems",
  "type": "peer-reviewed",
  "updated_at": "2026-10-31T00:20:46Z",
  "url": "https://research.draconitic.com/paper/aveqfd2maqx5v",
  "venue": "Mathematics",
  "versions": []
}