Geodesic
Asymptotic Solutions of the Cauchy Problem with Localized Initial Data for a Finite-Difference Scheme Corresponding to the One-Dimensional Wave Equation
Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 744-760

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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.
Keywords: wave equation, asymptotic solution, finite-difference scheme, nonstandard characteristics, Lagrangian manifold, vertical manifold. 
@article{MZM_2019_106_5_a8,
     author = {S. A. Sergeev},
     title = {Asymptotic {Solutions} of the {Cauchy} {Problem} with {Localized} {Initial} {Data} for a {Finite-Difference} {Scheme} {Corresponding} to the {One-Dimensional} {Wave} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {744--760},
     publisher = {mathdoc},
     volume = {106},
     number = {5},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a8/}
}
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S. A. Sergeev. Asymptotic Solutions of the Cauchy Problem with Localized Initial Data for a Finite-Difference Scheme Corresponding to the One-Dimensional Wave Equation. Matematičeskie zametki, Tome 106 (2019) no. 5, pp. 744-760. http://geodesic.mathdoc.fr/item/MZM_2019_106_5_a8/