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Volume 36, Issue 4
High Order Stable Multi-Domain Hybrid RKDG and WENO-FD Methods

Fan Zhang, Jian Cheng & Tiegang Liu

DOI: 10.4208/jcm.1702-m2016-0707

J. Comp. Math., 36 (2018), pp. 517-541.

Published online: 2018-06

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  • Abstract

Recently, a kind of high order hybrid methods based on Runge-Kutta discontinuous Galerkin (RKDG) method and weighted essentially non-oscillatory finite difference (WENO-FD) scheme was proposed. Those methods are computationally efficient, however, stable problems might sometimes be encountered in practical applications. In this work, we first analyze the linear stabilities of those methods based on the Heuristic theory. We find that the conservative hybrid method is linearly unstable if the numerical flux at the coupling interface is chosen to be 'downstream'. Then we introduce two ways of healing this defect. One is to choose the numerical flux at the coupling interface to be 'upstream'. The other is to employ a slope limiter function to enforce the hybrid method satisfying the local total variation diminishing (TVD) condition. In the end, numerical experiments are provided to validate the effectiveness of the proposed methods.

  • Keywords

Runge-Kutta discontinuous Galerkin method, Weighted essentially non-oscillatory scheme, Multi-domain hybrid method, Conservation laws, Heuristic theory.

  • AMS Subject Headings

65M60, 65M99, 35L65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhangfan1990@buaa.edu.cn (Fan Zhang)

chengjian@buaa.edu.cn (Jian Cheng)

liutg@buaa.edu.cn (Tiegang Liu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-517, author = {Zhang , FanCheng , Jian and Liu , Tiegang}, title = {High Order Stable Multi-Domain Hybrid RKDG and WENO-FD Methods}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {4}, pages = {517--541}, abstract = {

Recently, a kind of high order hybrid methods based on Runge-Kutta discontinuous Galerkin (RKDG) method and weighted essentially non-oscillatory finite difference (WENO-FD) scheme was proposed. Those methods are computationally efficient, however, stable problems might sometimes be encountered in practical applications. In this work, we first analyze the linear stabilities of those methods based on the Heuristic theory. We find that the conservative hybrid method is linearly unstable if the numerical flux at the coupling interface is chosen to be 'downstream'. Then we introduce two ways of healing this defect. One is to choose the numerical flux at the coupling interface to be 'upstream'. The other is to employ a slope limiter function to enforce the hybrid method satisfying the local total variation diminishing (TVD) condition. In the end, numerical experiments are provided to validate the effectiveness of the proposed methods.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1702-m2016-0707}, url = {http://global-sci.org/intro/article_detail/jcm/12303.html} }
TY - JOUR T1 - High Order Stable Multi-Domain Hybrid RKDG and WENO-FD Methods AU - Zhang , Fan AU - Cheng , Jian AU - Liu , Tiegang JO - Journal of Computational Mathematics VL - 4 SP - 517 EP - 541 PY - 2018 DA - 2018/06 SN - 36 DO - http://doi.org/10.4208/jcm.1702-m2016-0707 UR - https://global-sci.org/intro/article_detail/jcm/12303.html KW - Runge-Kutta discontinuous Galerkin method, Weighted essentially non-oscillatory scheme, Multi-domain hybrid method, Conservation laws, Heuristic theory. AB -

Recently, a kind of high order hybrid methods based on Runge-Kutta discontinuous Galerkin (RKDG) method and weighted essentially non-oscillatory finite difference (WENO-FD) scheme was proposed. Those methods are computationally efficient, however, stable problems might sometimes be encountered in practical applications. In this work, we first analyze the linear stabilities of those methods based on the Heuristic theory. We find that the conservative hybrid method is linearly unstable if the numerical flux at the coupling interface is chosen to be 'downstream'. Then we introduce two ways of healing this defect. One is to choose the numerical flux at the coupling interface to be 'upstream'. The other is to employ a slope limiter function to enforce the hybrid method satisfying the local total variation diminishing (TVD) condition. In the end, numerical experiments are provided to validate the effectiveness of the proposed methods.

Fan Zhang, Jian Cheng & Tiegang Liu. (2020). High Order Stable Multi-Domain Hybrid RKDG and WENO-FD Methods. Journal of Computational Mathematics. 36 (4). 517-541. doi:10.4208/jcm.1702-m2016-0707
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