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Volume 29, Issue 4
Effects of Integrations and Adaptivity for the Eulerian-Lagrangian Method

Jiwei Jia, Xiaozhe Hu, Jinchao Xu & Chen-Song Zhang

DOI: 10.4208/jcm.1012-m3397

J. Comp. Math., 29 (2011), pp. 367-395.

Published online: 2011-08

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  • Abstract

This paper provides an analysis on the effects of exact and inexact integrations on stability, convergence, numerical diffusion, and numerical oscillations for the Eulerian-Lagrangian method (ELM). In the finite element ELM, when more accurate integrations are used for the right-hand-side, less numerical diffusion is introduced and better approximation is obtained. When linear interpolation is used for numerical integrations, the resulting ELM is shown to be unconditionally stable and of first-order accuracy. When Gauss quadrature is used, conditional stability and second-order accuracy are established under some mild constraints for the convection-diffusion problems. Finally, numerical experiments demonstrate that more accurate integrations lead to better approximation, and spatial adaptivity can substantially reduce numerical oscillations and smearing that often occur in the ELM when inexact numerical integrations are used.

  • Keywords

Convection-diffusion problems, Eulerian–Lagrangian method, Adaptive mesh refinement.

  • AMS Subject Headings

65M25, 65M60.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-29-367, author = {Jiwei Jia, Xiaozhe Hu, Jinchao Xu and Chen-Song Zhang}, title = {Effects of Integrations and Adaptivity for the Eulerian-Lagrangian Method}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {4}, pages = {367--395}, abstract = {

This paper provides an analysis on the effects of exact and inexact integrations on stability, convergence, numerical diffusion, and numerical oscillations for the Eulerian-Lagrangian method (ELM). In the finite element ELM, when more accurate integrations are used for the right-hand-side, less numerical diffusion is introduced and better approximation is obtained. When linear interpolation is used for numerical integrations, the resulting ELM is shown to be unconditionally stable and of first-order accuracy. When Gauss quadrature is used, conditional stability and second-order accuracy are established under some mild constraints for the convection-diffusion problems. Finally, numerical experiments demonstrate that more accurate integrations lead to better approximation, and spatial adaptivity can substantially reduce numerical oscillations and smearing that often occur in the ELM when inexact numerical integrations are used.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1012-m3397}, url = {http://global-sci.org/intro/article_detail/jcm/8484.html} }
TY - JOUR T1 - Effects of Integrations and Adaptivity for the Eulerian-Lagrangian Method AU - Jiwei Jia, Xiaozhe Hu, Jinchao Xu & Chen-Song Zhang JO - Journal of Computational Mathematics VL - 4 SP - 367 EP - 395 PY - 2011 DA - 2011/08 SN - 29 DO - http://doi.org/10.4208/jcm.1012-m3397 UR - https://global-sci.org/intro/article_detail/jcm/8484.html KW - Convection-diffusion problems, Eulerian–Lagrangian method, Adaptive mesh refinement. AB -

This paper provides an analysis on the effects of exact and inexact integrations on stability, convergence, numerical diffusion, and numerical oscillations for the Eulerian-Lagrangian method (ELM). In the finite element ELM, when more accurate integrations are used for the right-hand-side, less numerical diffusion is introduced and better approximation is obtained. When linear interpolation is used for numerical integrations, the resulting ELM is shown to be unconditionally stable and of first-order accuracy. When Gauss quadrature is used, conditional stability and second-order accuracy are established under some mild constraints for the convection-diffusion problems. Finally, numerical experiments demonstrate that more accurate integrations lead to better approximation, and spatial adaptivity can substantially reduce numerical oscillations and smearing that often occur in the ELM when inexact numerical integrations are used.

Jiwei Jia, Xiaozhe Hu, Jinchao Xu and Chen-Song Zhang. (2011). Effects of Integrations and Adaptivity for the Eulerian-Lagrangian Method. Journal of Computational Mathematics. 29 (4). 367-395. doi:10.4208/jcm.1012-m3397
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