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Volume 14, Issue 3
Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations with Singular Initial Data

Li Guo & Yang Yang

Int. J. Numer. Anal. Mod., 14 (2017), pp. 342-354.

Published online: 2017-06

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  • Abstract

In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree $k$ and suitable initial discretizations, the DG solution is $(2k+1)$-th order accurate at the downwind points and $(k+2)$-th order accurate at all the other downwind-biased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of $k+1$ at all the interior upwind-biased Radau points. Besides the above, the DG solution is also $(k+2)$-th order accurate towards a particular projection of the exact solution and the numerical cell averages are $(2k+1)$-th order accurate. Numerical experiments are presented to confirm the theoretical results.

  • Keywords

Discontinuous Galerkin (DG) method, singular initial data, linear hyperbolic equations, superconvergence, weight function, weighted norms.

  • AMS Subject Headings

65M15, 65M60, 65N15, 60N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-342, author = {Li Guo and Yang Yang}, title = {Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations with Singular Initial Data}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {3}, pages = {342--354}, abstract = {

In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree $k$ and suitable initial discretizations, the DG solution is $(2k+1)$-th order accurate at the downwind points and $(k+2)$-th order accurate at all the other downwind-biased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of $k+1$ at all the interior upwind-biased Radau points. Besides the above, the DG solution is also $(k+2)$-th order accurate towards a particular projection of the exact solution and the numerical cell averages are $(2k+1)$-th order accurate. Numerical experiments are presented to confirm the theoretical results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10011.html} }
TY - JOUR T1 - Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations with Singular Initial Data AU - Li Guo & Yang Yang JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 342 EP - 354 PY - 2017 DA - 2017/06 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10011.html KW - Discontinuous Galerkin (DG) method, singular initial data, linear hyperbolic equations, superconvergence, weight function, weighted norms. AB -

In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree $k$ and suitable initial discretizations, the DG solution is $(2k+1)$-th order accurate at the downwind points and $(k+2)$-th order accurate at all the other downwind-biased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of $k+1$ at all the interior upwind-biased Radau points. Besides the above, the DG solution is also $(k+2)$-th order accurate towards a particular projection of the exact solution and the numerical cell averages are $(2k+1)$-th order accurate. Numerical experiments are presented to confirm the theoretical results.

Li Guo and Yang Yang. (2017). Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations with Singular Initial Data. International Journal of Numerical Analysis and Modeling. 14 (3). 342-354. doi:
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