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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
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