An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations

By Ngoc Cuong Nguyen, Jaime Peraire, Bernardo Cockburn. Journal of Computational Physics, 2009

386 citations recorded by the source corpus.

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection-diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton-Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is (p+1)th order accurate and when polynomials of degree p>=0 are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p+1 in the L^2-norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p+1 in the L^2-norm. The new approximate scalar variable is shown to converge with order p+2 in the L^2-norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/698b82a0d13496b240811f42cf1ad841477ac7a9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Numerical Analysis

Citations in the corpus

A moving discontinuous Galerkin finite element method for flows with interfaces (2018, background): “The shock interface emanating from (x, t) = 0 is governed by the familiar Rankine-Hugoniot condition for one-dimensional unsteady flow, which dictates the shock speed, ie, vs = ⟦x(y)⟧ ⟦y⟧ = [[ 1 2 y2 ]] ⟦y⟧ = {{y}}, (65) where ⟦y⟧ = (y+−y−) and {{y}} = 1 2 (y++y−).”
Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier-Stokes equations ✩ (2014, neutral): “Some useful criteria are provided in [20, 23, 24].”
Efficiency of high‐order elements for continuous and discontinuous Galerkin methods (2013, positive): “hybridizable discontinuous Galerkin (HDG) [4, 5, 14, 13, 15] are compared based on runtime evaluations and estimates for number of DOF (ndof) and minimal upper bandwidth for two-dimensional (2D) structured meshes, which are indicators of the computer cost.”
Hybridizable discontinuous Galerkin p‐adaptivity for wave propagation problems (2013, methodology): “Recently, however, a novel DG approach has been proposed where every interior node is uncoupled from neighboring element nodes: the hybridizable discontinuous Galerkin method (HDG), see for example [35, 36, 37, 38].”
A high‐order hybridizable discontinuous Galerkin method for elliptic interface problems (2013, neutral): “The optimal value of (cid:8) turns out to be a unit length [18].”
High‐order continuous and discontinuous Galerkin methods for wave problems (2013, methodology): “Discontinuous Galerkin methods [25, 26, 27, 28, 29, 30] can also be an alternative in this area.”
A class of discontinuous Petrov–Galerkin methods. II. Optimal test functions (2011, neutral): “In particular, for our primary example of the convection dominated diffusion problem, DG methods have remained an active area of research [13, 15, 20, 21, 22, 23, 31].”