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.
- Pseudo-time integration
- Implicit–explicit Runge–Kutta methods
- Compressible Navier–Stokes equations
- Discontinuous Galerkin finite element methods
Klaij, C. M., van der Vegt, J. J. W., & van der Ven, H. (2006). Pseudo-time stepping methods for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. Journal of computational physics, 219(500-266), 622-643. [10.1016/j.jcp.2006.04.003]. https://doi.org/10.1016/j.jcp.2006.04.003