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Transparent boundary conditions and numerical computation for singularly perturbed telegraph equation on unbounded domain. (English) Zbl 1450.65124

The asymptotic behavior of the solution for the singularly perturbed telegraph equation is studied on an unbounded domain by matched asymptotic expansions. A reduced initial-boundary value problem is introduced by deriving the transparent boundary conditions (TBCs) at the artificial boundaries, which is equivalent to the original problem on the bounded computational domain. The reduced problem is solved using Crank-Nicolson Galerkin scheme incorporated with the exponential wave integrator technique. The uniform convergence rate of the Crank-Nicolson Galerkin scheme is found to be \({\mathcal O}(h+\tau^\gamma)\) with \(0.4 \le\gamma\le 1\). The reliability and efficiency of the TBCs are shown in the numerical examples and the convergence rates of the Crank-Nicolson Galerkin scheme are validated.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin 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
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35M13 Initial-boundary value problems for PDEs of mixed type
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