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 $$t=0$$ t = 0 . Next, we introduce the artificial boundaries $$\varGamma _{\pm }$$ Γ ± to get a finite computational domain $$\varOmega _0$$ Ω 0 and derive the transparent boundary conditions on $$\varGamma _{\pm }$$ Γ ± for SPTE. Hence, we can reduce the original problem to an initial-boundary value problem (IBVP) on the bounded domain $$\varOmega _0$$ Ω 0 , and then the equivalence between the original problem and the IBVP on $$\varOmega _0$$ Ω 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.

中文翻译：

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无界域奇异摄动电报方程的透明边界条件及数值计算

在本文中，我们研究了无界域上奇异摄动电报方程（SPTE）的数值解。首先，我们通过渐近分析研究了SPTE解的第一次一致有效渐近展开，得到SPTE的解在$$t=0$$t=0附近有一个初始层。接下来，我们引入人工边界 $$\varGamma _{\pm }$$ Γ ± 得到有限计算域 $$\varOmega _0$$ Ω 0 并推导出 $$\varGamma _{\pm 上的透明边界条件}$$ Γ ± 用于 SPTE。因此，我们可以将原始问题简化为有界域 $$\varOmega _0$$ Ω 0 上的初始边界值问题（IBVP），然后原始问题与 $$\varOmega _0$ 上的 IBVP 等价$Ω 0 得到证明。此外，我们提出了一个 Crank-Nicolson Galerkin 方案来解决简化问题。此外，我们使用指数波积分器方法来处理初始层。我们还分析了 Crank-Nicolson Galerkin 方案的稳定性和收敛性。最后，一些数值例子验证了我们的理论结果，并显示了透明边界条件和 Crank-Nicolson Galerkin 方案的效率和可靠性。