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

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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.

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Authors and Affiliations

  1. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, 100088, Beijing, China

    Wang Kong

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Zhongyi Huang

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  1. Wang Kong
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Correspondence to Zhongyi Huang.

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This work was partially supported by the National Key Research and Development Plan of China 2017YFC0601801, NSFC Project No. 11871298 and Tsinghua University Initiative Scientific Research Program.

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Kong, W., Huang, Z. Transparent boundary conditions and numerical computation for singularly perturbed telegraph equation on unbounded domain. Numer. Math. 145, 345–382 (2020). https://doi.org/10.1007/s00211-020-01115-1

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