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**An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients.**
*(English)*
Zbl 1070.65076

Summary: We propose a three level implicit unconditionally stable difference scheme of \(O(k^2+ h^2)\) for the difference solution of second-order linear hyperbolic equation \(u_{tt}+ 2\alpha(x, t)u_t+ \beta^2(x, t)u= A(x, t)u_{xx}+ f(x,t)\), \(0< x< 1\), \(t> 0\) subject to appropriate initial and Dirichlet boundary conditions, where \(A(x, t)> 0\), \(\alpha(x, t)>\beta(x, t)\geq 0\). The proposed formula is applicable to the problems having singularity at \(x= 0\). The resulting tri-diagonal linear system of equations is solved by using the Gauss-elimination method. Numerical examples are provided to illustrate the unconditionally stable character of the proposed method.

### MSC:

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

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

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

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

### Keywords:

linear hyperbolic equation; variable coefficients; implicit scheme; singular equation; telegraph equation; unconditional stability; RMS errors; finite difference; second-order; Gauss-elimination method; Numerical examples
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\textit{R. K. Mohanty}, Appl. Math. Comput. 165, No. 1, 229--236 (2005; Zbl 1070.65076)

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### References:

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