A Direct Discontinuous Galerkin Method with Interface Correction for the Compressible Navier-Stokes Equations on Unstructured Grids

Abstract

Since the original DDG method has been introduced by Liu et al. [8] in 2009, a variety of DDG type methods have been proposed and further developed. In this paper, we further investigate and develop a new DDG method with interface correction (DDG (IC)) as the discretization of viscous and heat fluxes for the compressible Navier-Stokes equations on unstructured grids. Compared to the original DDG method, the newly developed DDG (IC) method demonstrates its superior in delivering the optimal order of accuracy under demanding situations. Strategies in extension and application of this newly developed DDG (IC) method for solving the compressible Navier-Stokes equations and special treatments designed for handling boundary viscous fluxes are presented and examined in this work. The performance of the new DDG method with interface correction is carefully evaluated and assessed through a number of typical test cases. Numerical experiments show that the new DDG method with interface correction can achieve the optimal order of accuracy on both uniform structured grids and nonuniform unstructured grids, which clearly indicates its potential for further applications of real engineering practices.

A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

“…Regarding the viscous numerical flux, they employed a productrule approach that is consistent with our method on the continuous level. More recently, Yue et al [39] and Cheng et al [40] extended the first DDG method to have more interface terms added and developed symmetric and interface correction versions of DDG method for NS equations, respectively. Their method might be thought of as a generalization of the IPDG method of Hartmann and Houston [25] by including jump terms for second order derivatives.…”

Meshfree lattice Boltzmann flux solver for compressible inviscid flows

“…Structured grids normally discretize the computational domain with quadrilateral or hexahedral elements, which is difficult to describe complex geometries. Although unstructured grids can better adapt to the complex geometries 2,3 than structured grids and reduce the difficulty of grid generation, it is still a heavy task to generate satisfactory grids. Therefore, the grid generation is the bottleneck of the development and application of CFD.…”