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**High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation.**
*(English)*
Zbl 1156.65087

Summary: We introduce a high-order accurate method for solving a two dimensional linear hyperbolic equation. We apply a compact finite difference approximation of fourth order for discretizing the spatial derivatives of the linear hyperbolic equation and a collocation method for the time component. The resulted method is unconditionally stable and solves the two-dimensional linear hyperbolic equation with high accuracy. In this technique, the solution is approximated by a polynomial at each grid point the coefficients of which are determined by solving a system of linear equations. Numerical results show that the compact finite difference approximation of fourth order and the collocation method give a very efficient approach for solving the two dimensional linear hyperbolic equation.

### MSC:

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

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

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

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

### Keywords:

collocation technique; compact finite difference scheme; high accuracy; linear hyperbolic equation; stability; numerical results
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\textit{M. Dehghan} and \textit{A. Mohebbi}, Numer. Methods Partial Differ. Equations 25, No. 1, 232--243 (2009; Zbl 1156.65087)

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### References:

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