Numerical viscosity in large time step hll-type schemes

We consider Large Time Step (LTS) methods, i.e., the explicit finite volume methods not limited by the Courant–Friedrichs–Lewy (CFL) condition. The original LTS method (LeVeque in SIAM J Numer Anal 22, 1985) was constructed as an extension of the Godunov scheme, and successive versions have been developed in the framework of Roe’s approximate Riemann solver. Recently, Prebeg et al. (in ESAIM: M2AN, in press, 2017) developed the LTS extension of the HLL and HLLC schemes. We perform the modified equation analysis and demonstrate that for the appropriate choice of the wave velocity estimates, the LTS-HLL scheme yields entropy-satisfying solutions. We apply the LTS-HLL(C) schemes to the one-dimensional Euler equations and consider the Sod shock tube, double rarefaction, and Woodward–Colella blast-wave problem