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On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations

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  • Hendrik Ranocha

    (University of Münster)

  • Manuel Quezada Luna

    (King Abdullah University of Science and Technology (KAUST))

  • David I. Ketcheson

    (King Abdullah University of Science and Technology (KAUST))

Abstract

We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.

Suggested Citation

  • Hendrik Ranocha & Manuel Quezada Luna & David I. Ketcheson, 2021. "On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations," Partial Differential Equations and Applications, Springer, vol. 2(6), pages 1-26, December.
    • Handle: RePEc:spr:pardea:v:2:y:2021:i:6:d:10.1007_s42985-021-00126-3
    DOI: 10.1007/s42985-021-00126-3
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    References listed on IDEAS

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    1. Calvo, M. & Laburta, M.P. & Montijano, J.I. & Rández, L., 2011. "Error growth in the numerical integration of periodic orbits," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(12), pages 2646-2661.
    2. Gassner, Gregor J. & Winters, Andrew R. & Kopriva, David A., 2016. "A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 291-308.
    3. E. Momoniat, 2014. "A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-14, April.
    4. Kalisch, Henrik & Lenells, Jonatan, 2005. "Numerical study of traveling-wave solutions for the Camassa–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 25(2), pages 287-298.
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