Characteristic-Based Flux Splitting for Implicit-Explicit Time Integration of Low-Mach Number Flows
Low-Mach number flows, such as atmospheric flows, are characterized by two different time scales corresponding to the flow velocity and the acoustic waves. The stability limit of explicit time integration methods is restricted by the speed of sound; however, the acoustic waves often do not have a significant impact on the flow phenomenon. Most numerical algorithms for unsteady low-Mach number flows solve the Navier-Stokes equations with the incompressible assumption (where the speed of sound is assumed to be infinite), or use a preconditioned implicit time-integration method to integrate the compressible Navier-Stokes equations. In the context of atmospheric flows, an alternative approach is to eliminate the acoustic mode by using a hydrostatic model. Recently, implicit-explicit time-integration methods were applied to non-hydrostatic atmospheric flows (Giraldo, Restelli, Laeuter, SIAM J. Sci. Comput., 2010, and Giraldo, Kelly, Constantinescu, SIAM J. Sci. Comput., 2013) where a perturbation-based splitting of the hyperbolic flux was used. In this study, we propose a characteristic-based splitting of the convective flux in the Euler/Navier-Stokes equations to separate the velocity and acoustic time scales. The eigen-structure of the flux is used to decompose it into a “slow” flux consisting of the entropy wave propagating at the local flow velocity, and a “fast” flux consisting of the acoustic waves propagating at the relative speed of sound. Additive Runge-Kutta methods are used to integrate the former in time explicitly and the latter implicitly. Thus, the linear stability limit of the algorithm depends on the flow velocity, and not the speed of sound, while maintaining high order accuracy in time. This approach is used in a conservative finite-difference algorithm where the weighted essentially non-oscillatory (WENO) (Jiang, Shu, J. Comput. Phys., 1996) and the compact-reconstruction WENO (Ghosh, Baeder, SIAM J. Sci. Comput., 2012) schemes are used for the spatial discretization. We also propose a linearization of the “fast” flux in time such that the system of equations resulting from semi-implicit time integration can be solved efficiently without compromising the stability limit of the overall algorithm or introducing an error in the discretization. Our proposed approach is verified for several benchmark flow problems where the flow velocities are significantly smaller than the speed of sound. We present results that demonstrate the performance and accuracy of our algorithm.