The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections
Hailiang Liu 1*, Jue Yan 11 Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
Received 1 September 2009; Accepted (in revised version) 1 December 2009
Available online 15 April 2010
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 ($k \geq 4$) 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 $p^k$ elements. 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.
AMS subject classifications: 35K05, 35K15, 65N12, 65N30
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Key words: Diffusion, discontinuous Galerkin methods, stability, numerical flux.
Email: email@example.com (H. Liu), firstname.lastname@example.org (J. Yan)