Abstract
In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.
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References
Vishik, M.I. and L.A. Lyusternik, On the asymptotic behavior of the solution of boundary value problems for quasilinear differential equations,Dokl. Akad. Nauk., SSSR,121 (1958).
Vishik, M.I. and L.A. Lyusternik, Initial jump for nonlinear differential equations containing a small parameter,Sov. Math. Dokl.,1 (1960).
Kasymov, K.A., On the asymptotic solution of the Cauchy problem with large initial conditions for nonlinear ordinary differential equations containing a small parameter,Usp. Math. Nauk.,17, 5 (1962).
Kasymov, K. A., The initial jump problem for nonlinear system of differential equations containing a small parameter,Sov. Math. Dokl.,9 (1968).
Kasymov, K. A., Asymptotic solution of the Cauchy problem with initial jump for the second-order hyperbolic equation involving a small paramter,Sov. Math. Dokl.,195, 1 (1970).
Kasymov, K. A. and B. M. Kadykenov, On the Cauchy problem with initial jump for the singular perturbation linear hyperbolic equations degenerated to first order equation,Diff. Equations,19, 12 (1983).
Hsiao, G.C. and R. J. Weinacht, A singularly perturbed Cauchy problem,J. of Math. Anal. and Appl.,71 (1979).
Hsiao, G. C. and R. J. Weinacht, Singular perturbations for a semilinear hyperbolic equation,SIAM J. Math. Anal.,14, 6 (1983).
Pecenkina, A.A., The solution of periodic problem for the second order ordinary differential equations with a small parameter multiplying the highest derivative,Diff. Equations with a Small Parameter, SSSR Akademy of Science, Ural'skii Scientific Centre, Sverdlovsk (1980).
Su Yu-cheng and Lin Ping, Numerical solution of singular perturbation problem for hyperbolic differential equation with characteristic boundaries,proceedings of the BAIL V Conference, Shanghai, June (1988).
Doolan,E.P., J. J. H. Miller and W.H.A. Schilders,Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin (1980)
Lees, M., Energy inequalities for the solution of differential equations,Trans. Amer. Math. Soc.,94, 1 (1960).
Shishkin, G. I., Numerical solution of the boundary value problem for elliptic equations with small parameter at the leading derivative,USSR Computational Math. Math. Phys.,26, 7–8 (1986).
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Yu-cheng, S., Ping, L. Numerical solution of the singularly perturbed problem for the hyperbolic equation with initial jump. Appl Math Mech 11, 709–721 (1990). https://doi.org/10.1007/BF02015145
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DOI: https://doi.org/10.1007/BF02015145