High-resolution large time-step schemes for inviscid fluid flow

•A full set of conditions for high-resolution LTS is presented.•A high-resolution version (second order TVD) of the LTS-Roe scheme is derived.•The second-order Large Time Step scheme is compared to the high-resolution Large Time Step scheme proposed by Harten. We present a set of sufficient conditions for conservative, consistent and total variation diminishing (TVD) Large Time Step (LTS) schemes. We generalize the modified flux approach of Harten (1986) to achieve second-order accuracy away from discontinuities. In particular, we derive a supplementary condition, providing a full set of sufficient conditions for high-resolution LTS. In this framework, a second-order extension (LTS-Roe2) of the LTS-Roe scheme is proposed and compared to the second order LTS scheme proposed by Harten. Tests show that LTS-Harten consistently gives good results with less oscillations than LTS-Roe2, but has a tendency to smear out discontinuities when the Courant number is increased.