An Asymptotic-Preserving all-speed scheme for the Euler and Navier–Stokes equations

We present an Asymptotic-Preserving ‘all-speed’ scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and two-dimensional numerical results provide a validation of the Asymptotic-Preserving ‘all-speed’ properties.