Abstract
We pose the Cauchy problem with localized initial data that arises when passing from an explicit difference scheme for the wave equation to a pseudodifferential equation. The solution of the Cauchy problem for the difference scheme is compared with the asymptotics of the solution of the Cauchy problem for the pseudodifferential equation. We give a detailed study of the behavior of the asymptotic solution in the vicinity of the leading edge, where yet another version of the asymptotic solution is constructed based on vertical manifolds.
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Acknowledgments
The author wishes to express gratitude to S. A. Goreinov, V. G. Danilov, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov, and A. V. Tsvetkova for useful discussions and remarks.
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This work was supported by the Russian Science Foundation under grant 16-11-10282.
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 5, pp. 744–760.
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Sergeev, S.A. Asymptotic Solutions of the Cauchy Problem with Localized Initial Data for a Finite-Difference Scheme Corresponding to the One-Dimensional Wave Equation. Math Notes 106, 800–813 (2019). https://doi.org/10.1134/S0001434619110130
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DOI: https://doi.org/10.1134/S0001434619110130