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Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

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Abstract

In this paper, we present and analyze an arbitrary Lagrangian–Eulerian local discontinuous Galerkin (ALE-LDG) method for one-dimensional linear convection–diffusion problems. The semi-discrete ALE-LDG method is shown to preserve \(L^{2}\)-stability and sub-optimal (\(k+\frac{1}{2}\)) convergence rate, when piecewise polynomials of degree k on the reference cell are used and Lax–Friedrichs flux is taken for the convection term. In addition, we also discuss three specific fully discrete ALE-LDG schemes, in which implicit–explicit Runge–Kutta (IMEX) time-marching is applied. With the aid of scaling arguments and the standard energy analysis, we prove that the corresponding fully discrete schemes are stable provided the time step \(\tau \le \tau _{0}\), where the positive constant \(\tau _{0}\) is independent of the mesh size h but depends on the convection and diffusion coefficients, the polynomial degree, and the moving grid function. Under the time step restriction, we obtain quasi-optimal error estimate in space and optimal convergence rate in time for the fully discrete schemes. Numerical examples are also given to illustrate our theoretical results.

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The datasets generated during the current study are available from the corresponding author on reasonable request. They support our published claims and comply with field standards.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, Henan, People’s Republic of China

    Lingling Zhou

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Yinhua Xia

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L. Zhou: Research of this author is supported by NSFC Grant No. 12001171.

Y. Xia: Research supported by the National Numerical Windtunnel Project NNW2019ZT4-B08 and a NSFC Grant No. 11871449.

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Zhou, L., Xia, Y. Arbitrary Lagrangian–Eulerian Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations. J Sci Comput 90, 21 (2022). https://doi.org/10.1007/s10915-021-01697-4

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