The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems

Abstract

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), . 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 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.

A local discontinuous Galerkin method for the Burgers–Poisson equation

“…The idea of the LDG method is to rewrite higher order equations into a first order system, and then apply the DG method on the system. In contrast, the direct discontinuous Galerkin (DDG) methods, proposed in [26,27] primarily for diffusion equations, aimed at directly solving higher order PDEs by the DG discretization, see e.g., [2,40] for energy preserving DG methods for KdV type equations, and [24] for the Degasperis-Procesi equation. The DDG method, as another class of DG methods for higher order partial differential equations, is to directly force the weak solution formulation of the PDE into the DG function space for both the numerical solution and test functions.…”

“…Unlike the traditional LDG method, the DDG method does not rewrite the original equation into a larger first order system. The main novelty in the DDG schemes proposed in [26,27] lies in numerical flux choices for the solution gradient, which involves higher order derivatives evaluated crossing cell interfaces.…”

Discontinuous Legendre Wavelet Galerkin Method for One-Dimensional Advection-Diffusion Equation

“…Finally, the DLWG approach utilizes the discontinuous feature at nodes of the Legendre wavelet bases combined with discontinuous finite elements to discretize the space variable and the spacial derivatives to produce a system of first-order ODEs in time for Equation (1.1). We solve this system by using the TVD Runge-Kutta method [11], and obtain good numerical results, illustrating that this scheme is very simple and computationally efficient. This paper is organized as follows: in Section 2, descriptions of the Legendre wavelet and its rich properties are given.…”

“…However, numerical experience suggests that as the degree k of the approximate solution increases, the choice of the numerical fluxes does not have significant impact on the quality of the approximations [11].…”

Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations

“…For the evaluation of the viscous fluxes the extrapolated interface variables U ± i,a and their unlimited gradients ∇U ± i,a from the k-exact least square reconstruction of equation (12)a r e averaged from two discontinuous states as detailed in [3,47]. Although, other methods for the evaluation of the diffusive fluxes such as the one of the generalised Riemann problem of Gassner et al [48], or the diffusive flux of direct DG [49,50]c a nb ea p p l i e d ,t h ek -o r d e ra c c u r a t efl u xi si m p l e m e n t e da si te a s i l yo b t a i n e d through element centred reconstructions and has been applied for various flows problems [47,51,52].…”