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Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Author

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  • Isaías Alonso-Mallo

    (Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Valladolid, Paseo de Belén 7, 47011 Valladolid, Spain
    Instituto de Investigación en Matemáticas de la Universidad de Valladolid (IMUVa), 47011 Valladolid, Spain.
    These authors contributed equally to this work.)

  • Ana M. Portillo

    (Departamento de Matemática Aplicada, Escuela de Ingenierías Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain
    Instituto de Investigación en Matemáticas de la Universidad de Valladolid (IMUVa), 47011 Valladolid, Spain.
    These authors contributed equally to this work.)

Abstract

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

Suggested Citation

  • Isaías Alonso-Mallo & Ana M. Portillo, 2021. "Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods," Mathematics, MDPI, vol. 9(10), pages 1-24, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1113-:d:554694
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    References listed on IDEAS

    as
    1. Portillo, A.M., 2018. "High-order full discretization for anisotropic wave equations," Applied Mathematics and Computation, Elsevier, vol. 323(C), pages 1-16.
    2. Cano, B. & Moreta, M.J., 2020. "A modified Gautschi’s method without order reduction when integrating boundary value nonlinear wave problems," Applied Mathematics and Computation, Elsevier, vol. 373(C).
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