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Analysis of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional Nonlinear Fourth-Order Time-Dependent Problems

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Abstract

In this paper, we study the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear time-dependent fourth-order equations. The \(L^2\) stability and optimal error estimates of order \(k+1\) are obtained with piecewise polynomials of degree \(k \ge 1\). The numerical flux of the nonlinear convection term is the generalized local Lax–Friedrichs flux, and the generalized alternating fluxes are adopted for the fourth- and second-order terms. The adjustable numerical viscosity of these fluxes is beneficial for long time simulations with a slower error growth. A proper numerical initial condition is designed based on a modified projection of the third-order derivative. By using the generalized Gauss–Radau projections, together with sharp bounds of nonlinear and jump terms, optimal error estimates are derived. The results are extended to equations with an additional dispersion term and mixed boundary conditions. Examples including Dirichlet as well as generalized Dirichlet boundary conditions, singularly perturbed problems and two-dimensional problems are also numerically investigated, indicating that the theoretical results hold for more general cases.

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Data Availability

No datasets were generated or analyzed during the current study. The codes required during the algorithm implementation are available from the corresponding author on reasonable request.

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Acknowledgements

The authors are grateful to anonymous reviewers for their valuable suggestions, which greatly improved the quality of the paper.

Funding

Xiong Meng was supported by NSFC Grants 12371365, 11971132, Natural Science Foundation of Heilongjiang Province Grant YQ2021A002, and the Fundamental Research Funds for the Central Universities Grant HIT.OCEF.2022031. Jia Li was supported by NSFC Grant 12101158 and the Fellowship of China Postdoctoral Science Foundation Grant 2021M701010.

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  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, China

    Linhui Li, Xiong Meng & Jia Li

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Li, L., Meng, X. & Li, J. Analysis of the Local Discontinuous Galerkin Method with Generalized Numerical Fluxes for One-Dimensional Nonlinear Fourth-Order Time-Dependent Problems. J Sci Comput 102, 72 (2025). https://doi.org/10.1007/s10915-025-02800-9

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