# Interdisciplinary Applied Mathematics

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sion of the spectral element discretization combined with the characteristic approach for the boundary conditions.

Turning now to the full compressible Navier-Stokes equations, we need to treat properly the viscous fluxes at elemental interfaces as well as the noslip or    slip    condition.    First,    unlike the    spatial    discretization    for    the    Euler

equations, where we employ a spectral collocation approach, for the viscous contributions we employ a standard Galerkin projection (Karniadakis and Sherwin, 1999). Specifically, the viscous diffusion terms in the momentum equation are treated by the chain rule of differentiation, because the unknowns are the conservative variables, not the primitive variables, i.e., (pu, pv,E), not (u,v,T). The Laplacian of temperature is treated by assuming that T is also an independent variable. This seems necessary; otherwise, the spatial derivatives of T must include spatial derivatives of terms obtained from the chain rule of differentiation of the energy relation (E = p[T + 1/2(u2 + v2)]). The elemental interface is treated by applying a direct stiffness summation procedure as in standard finite element methods (Karniadakis and Sherwin, 1999). Specifically, simple addition of all contributions at nodes at the interface is performed to ensure continuity of the variables.

No-slip and Dirichlet temperature boundary conditions are implemented at the walls. It is possible to specify Dirichlet boundary conditions at the inflow and outflow, which can be a function of time and space. Use of the characteristic treatment is essential for the stability of the inviscid part of the equations, and viscous boundary conditions must be designed to maintain minimal wave reflection from the outflow and inflow boundaries. At the outflow we let the pressure be infinitesimally smaller than the value calculated from the inviscid step. This treatment is usually enough to release the pressure waves with minimal reflection from the boundaries for the viscous substep.

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