Pseudo-time stepping methods for space–time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

• Published in: Journal of computational physics Vol. 219; no. 2; pp. 622 - 643
• Main Authors: Klaij, C.M., van der Vegt, J.J.W., van der Ven, H.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.12.2006; Elsevier
• Subjects: Airfoils; Compressible Navier–Stokes equations; Computation; Computational techniques; Discontinuous Galerkin finite element methods; Discretization; Exact sciences and technology; Galerkin methods; Implicit–explicit Runge–Kutta methods; Mathematical analysis; Mathematical methods in physics; Navier-Stokes equations; Physics; Pseudo-time integration; Stability; Three dimensional; 02.70.Dh; Pseudo-time integration; Compressible Navier–Stokes equations; Implicit–explicit Runge–Kutta methods; 02.60.Cb; 03.40.Gc; Discontinuous Galerkin finite element methods; Reynolds number; Euler equations; Two dimensional flow; Calculation methods; Galerkin-Petrov method; Finite element method; Space-time; Discretization; Algebraic equation; Calculation; Runge-Kutta methods; Performance; Nonlinear systems; Navier-Stokes equations; Compressible Navier-Stokes equations; Implicit- explicit Runge-Kutta methods
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2006.04.003

The space–time discontinuous Galerkin discretization of the compressible Navier–Stokes equations results in a non-linear system of algebraic equations, which we solve with pseudo-time stepping methods. We show that explicit Runge–Kutta methods developed for the Euler equations suffer from a severe stability constraint linked to the viscous part of the equations and propose an alternative to relieve this constraint while preserving locality. To evaluate its effectiveness, we compare with an implicit–explicit Runge–Kutta method which does not suffer from the viscous stability constraint. We analyze the stability of the methods and illustrate their performance by computing the flow around a 2D airfoil and a 3D delta wing at low and moderate Reynolds numbers.