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.


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
Full Text: DOI


[1] Twizell, E.H., An explicit difference method for the wave equation with extended stability range, Bit, 19, 378-383, (1979) · Zbl 0441.65066
[2] Mohanty, R.K.; Jain, M.K.; George, K., On the use of high order difference methods for the system of one space second order non-linear hyperbolic equations with variable coefficients, J. comp. appl. math., 72, 421-431, (1996) · Zbl 0877.65066
[3] Ciment, M.; Leventhal, S.H., A note on the operator compact implicit method for the wave equation, Mathematics of computation, 32, 143-147, (1978) · Zbl 0373.35039
[4] Mohanty, R.K.; Jain, M.K., An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Numer. meth. partial diff. eq., 17, 684-688, (2001) · Zbl 0990.65101
[5] Mohanty, R.K.; Jain, M.K.; Arora, U., An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensions, Int. J. computer math., 79, 133-142, (2002) · Zbl 0995.65093
[6] Debnath, L., Nonlinear partial differential equations for scientists and engineers, (1997), Birkhauser Boston, MA · Zbl 0892.35001
[7] Dahlquist, G., On accuracy and unconditional stability of linear multi-step methods for second order differential equations, Bit, 18, 133-136, (1978) · Zbl 0378.65043
[8] Jain, M.K., Numerical solution of differential equations, (1984), Wiley Eastern · Zbl 0536.65004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.