TY - JOUR
T1 - A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations
AU - Shahbazi, Khosro
AU - Fischer, Paul F.
AU - Ethier, C. Ross
N1 - Funding Information:
The authors thank the anonymous reviewers for their valuable comments. This work was financially supported by the National Sciences and Engineering Council of Canada through operating grant A2191 and by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, US Department of Energy, under Contract W-31-109-Eng-38.
PY - 2007/3/1
Y1 - 2007/3/1
N2 - We present a high-order discontinuous Galerkin discretization of the unsteady incompressible Navier-Stokes equations in convection-dominated flows using triangular and tetrahedral meshes. The scheme is based on a semi-explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the Stokes operator. The nonlinear term is discretized in divergence form by using the local Lax-Friedrichs fluxes; thus, local conservativity is inherent. Spatial discretization of the Stokes operator has employed both equal-order (Pk - Pk) and mixed-order (Pk - Pk-1) velocity and pressure approximations. A second-order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic Helmholtz equation for each component of the velocity and a consistent Poisson equation for the pressure. The consistent Poisson operator is replaced by an equivalent (in stability and convergence) operator, namely that arising from the interior penalty discretization of the standard Poisson operator with appropriate boundary conditions. This yields a simpler and more efficient method, characterized by a compact stencil size. We show the temporal and spatial behavior of the method by solving some popular benchmarking tests. For an unsteady Stokes problem, second-order temporal convergence is obtained, while for the Taylor vortex test problem on both semi-structured and fully unstructured triangular meshes, spectral convergence with respect to the polynomial degree k is obtained. By studying the Orr-Sommerfeld stability problem, we demonstrate that the Pk - Pk method yields a stable solution, while the Pk - Pk-1 formulation leads to unphysical instability. The good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder. We conclude that the proposed discontinuous Galerkin method with the Pk - Pk formulation is an efficient scheme for accurate and stable solution of the unsteady Navier-Stokes equations in convection-dominated flows.
AB - We present a high-order discontinuous Galerkin discretization of the unsteady incompressible Navier-Stokes equations in convection-dominated flows using triangular and tetrahedral meshes. The scheme is based on a semi-explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the Stokes operator. The nonlinear term is discretized in divergence form by using the local Lax-Friedrichs fluxes; thus, local conservativity is inherent. Spatial discretization of the Stokes operator has employed both equal-order (Pk - Pk) and mixed-order (Pk - Pk-1) velocity and pressure approximations. A second-order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic Helmholtz equation for each component of the velocity and a consistent Poisson equation for the pressure. The consistent Poisson operator is replaced by an equivalent (in stability and convergence) operator, namely that arising from the interior penalty discretization of the standard Poisson operator with appropriate boundary conditions. This yields a simpler and more efficient method, characterized by a compact stencil size. We show the temporal and spatial behavior of the method by solving some popular benchmarking tests. For an unsteady Stokes problem, second-order temporal convergence is obtained, while for the Taylor vortex test problem on both semi-structured and fully unstructured triangular meshes, spectral convergence with respect to the polynomial degree k is obtained. By studying the Orr-Sommerfeld stability problem, we demonstrate that the Pk - Pk method yields a stable solution, while the Pk - Pk-1 formulation leads to unphysical instability. The good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder. We conclude that the proposed discontinuous Galerkin method with the Pk - Pk formulation is an efficient scheme for accurate and stable solution of the unsteady Navier-Stokes equations in convection-dominated flows.
KW - Algebraic splitting methods
KW - Discontinuous Galerkin methods
KW - High-order methods
KW - Incompressible flows
KW - Navier-Stokes equations
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U2 - 10.1016/j.jcp.2006.07.029
DO - 10.1016/j.jcp.2006.07.029
M3 - Article
AN - SCOPUS:33846837197
SN - 0021-9991
VL - 222
SP - 391
EP - 407
JO - Journal of Computational Physics
JF - Journal of Computational Physics
IS - 1
ER -