Positivity-preserving method for multi-resolution simulations of compressible flows

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.