Stability and Error Analysis of Multistep IMEX Local Discontinuous Galerkin Methods for One-Dimensional Nonlinear Fourth-Order Evolution Problems
- 1Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia
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.
Related
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BibTeX
@article{alharbi2026stability,
title = {{Stability and Error Analysis of Multistep IMEX Local Discontinuous Galerkin Methods for One-Dimensional Nonlinear Fourth-Order Evolution Problems}},
author = {Amjad G. Alharbi and Tarek Aboelenen},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213950},
url = {https://www.mdpi.com/2227-7390/14/21/3950},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Stability and Error Analysis of Multistep IMEX Local Discontinuous Galerkin Methods for One-Dimensional Nonlinear Fourth-Order Evolution Problems
AU - Amjad G. Alharbi
AU - Tarek Aboelenen
PY - 2026
DA - 2026/10/29
JO - Mathematics
DO - 10.3390/math14213950
UR - https://www.mdpi.com/2227-7390/14/21/3950
AB - 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.
SN - 2227-7390
ER -