Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

In this work, we consider the development of implicit-explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The proposed scheme is asymptotically stable with a CFL condition independent of the Mach number. In addition, it degenerates, in the low Mach number regime, to a consistent discretization of the incompressible system. Since it has been proved that implicit schemes of order higher than one cannot be TVD (SSP) [30], we construct a new paradigm of implicit time integrators by coupling first-order in time schemes with second-order ones in the same spirit as highly accurate shock-capturing TVD methods in space. For this particular class of schemes, the TVD property is first proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first- to the second-order both in space and time. It preserves the monotonicity of the solution, and is highly accurate for all choices of the Mach number. Moreover, the time step is only restricted by the non-stiff part of the system. We finally show, thanks to one- and two-dimensional test cases, that the method indeed possesses the claimed properties.