Transparent boundary conditions and numerical computation for singularly perturbed telegraph equation on unbounded domain

Abstract

In this paper, we study the numerical solution for the singularly perturbed telegraph equation (SPTE) on unbounded domain. Firstly, we investigate the first consistent effective asymptotic expansion for the solution of SPTE by the asymptotic analysis and obtain that the solutions of SPTE have an initial layer near $$t=0$$. Next, we introduce the artificial boundaries $$\varGamma _{\pm }$$ to get a finite computational domain $$\varOmega _0$$ and derive the transparent boundary conditions on $$\varGamma _{\pm }$$ for SPTE. Hence, we can reduce the original problem to an initial-boundary value problem (IBVP) on the bounded domain $$\varOmega _0$$, and then the equivalence between the original problem and the IBVP on $$\varOmega _0$$ is proved. In addition, we propose a Crank–Nicolson Galerkin scheme to solve the reduced problem. Furthermore, we use the exponential wave integrator method to deal with the initial layer. We also analyze the stability and convergence of the Crank–Nicolson Galerkin scheme. Finally, some numerical examples validate our theoretical results and show the efficiency and reliability of the transparent boundary conditions and the Crank–Nicolson Galerkin scheme.