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**An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation.**
*(English)*
Zbl 1046.65076

Summary: An implicit three-level difference scheme of \(O(k^2+h^2)\) is discussed for the numerical solution of the linear hyperbolic equation \(u_{tt}+2\alpha u_t+\beta^2u= u_{xx}+f(x, t),\;\alpha>\beta\geq 0\), in the region \(\Omega =\{(x,t)\mid 0<x<1,\;t>0\}\) subject to appropriate initial and Dirichlet boundary conditions, where \(\alpha\) and \(\beta\) are real numbers. We have used nine grid points with a single computational cell. The proposed scheme is unconditionally stable. The resulting system of algebraic equations is solved by using a tridiagonal solver. Numerical results demonstrate the required accuracy of the proposed scheme.

### MSC:

65M12 | Stability and convergence of numerical 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 |

### Keywords:

second-order linear hyperbolic equation; damped wave equation; telegraph equation; Padé approximation; unconditional stability; implicit three-level difference scheme; numerical results
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\textit{R. K. Mohanty}, Appl. Math. Lett. 17, No. 1, 101--105 (2004; Zbl 1046.65076)

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### References:

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