Volume 8, Issue 3
The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections

Hailiang Liu & Jue Yan

Commun. Comput. Phys., 8 (2010), pp. 541-564.

Published online: 2010-08

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  • Abstract

Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475-698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all pelements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one- and two-dimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.

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@Article{CiCP-8-541, author = {}, title = {The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections}, journal = {Communications in Computational Physics}, year = {2010}, volume = {8}, number = {3}, pages = {541--564}, abstract = {

Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475-698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all pelements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one- and two-dimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.010909.011209a}, url = {http://global-sci.org/intro/article_detail/cicp/7584.html} }
TY - JOUR T1 - The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections JO - Communications in Computational Physics VL - 3 SP - 541 EP - 564 PY - 2010 DA - 2010/08 SN - 8 DO - http://dor.org/10.4208/cicp.010909.011209a UR - https://global-sci.org/intro/article_detail/cicp/7584.html KW - AB -

Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475-698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all pelements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one- and two-dimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.

Hailiang Liu & Jue Yan. (2020). The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections. Communications in Computational Physics. 8 (3). 541-564. doi:10.4208/cicp.010909.011209a
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