  We present a positivity-preserving method for multi-resolution simulations of
compressible flows involving extreme conditions such as near vacuum and strong
discontinuities. The novelty of this work is due to two aspects. First we
extend the positivity-preserving flux limiter of (Hu et al., J Comput Phys 242,
2013) to the multi-resolution framework by modifying the prediction operator,
based on the same limiting concept. Second, we develop a positivity-preserving
local time stepping scheme for adaptive time marching. Instead of using fixed
hierarchical time steps, the local time stepping scheme dynamically adjusts the
time steps of all multi-resolution levels to maintain positivity. The method is
validated and its capabilities are demonstrated by a range of test cases.
