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.


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
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