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**High order compact solution of the one-space-dimensional linear hyperbolic equation.**
*(English)*
Zbl 1151.65071

Summary: We introduce a high-order accurate method for solving one-space dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing spatial derivative of linear hyperbolic equation and collocation method for the time component. The main property of this method additional to its high-order accuracy due to the fourth order discretization of spatial derivative, is its unconditionally stability. In this technique the solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations.

Numerical results show that the compact finite difference approximation of fourth order and collocation method produce a very efficient method for solving the one-space-dimensional linear hyperbolic equation. We compare the numerical results of this paper with numerical results of R. K. Mohanty [Appl. Math. Lett. 17, No. 1, 101–105 (2004; Zbl 1046.65076)].

Numerical results show that the compact finite difference approximation of fourth order and collocation method produce a very efficient method for solving the one-space-dimensional linear hyperbolic equation. We compare the numerical results of this paper with numerical results of R. K. Mohanty [Appl. Math. Lett. 17, No. 1, 101–105 (2004; Zbl 1046.65076)].

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35L15 | Initial value problems for second-order hyperbolic equations |

### Keywords:

collocation; compact finite difference scheme; linear hyperbolic equation; telegraph equation; high accuracy### Citations:

Zbl 1046.65076
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\textit{A. Mohebbi} and \textit{M. Dehghan}, Numer. Methods Partial Differ. Equations 24, No. 5, 1222--1235 (2008; Zbl 1151.65071)

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### References:

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