Positivity-preserving implicit finite volume methods on unstructured grids for compressible flows

Preserving the positivity of density and pressure remains a critical challenge in numerical simulations of complex compressible flows, particularly for methods with implicit time integration, where solutions are obtained through iterative nonlinear solves. In this work, we propose an efficient positivity-preserving algorithm for implicit high-order finite volume methods with dual-time stepping on unstructured grids. This algorithm enforces the positivity of cell-averaged solutions throughout the implicit time integration via two procedures. First, the physical-time step is adaptively limited to guarantee the existence of an admissible solution at the next time level. The future state is estimated via a linear approximation using the available semi-discrete finite volume residual and constrained by a positivity-preserving lower bound that accounts for estimation errors, to determine an allowable time step. Second, pseudo-time step limiting combined with increment correction is applied to maintain positivity of intermediate states during inner iterations. Given positive cell averages, admissible reconstruction polynomials are obtained using a scaling limiter. Importantly, the positivity-preserving algorithm is accuracy-preserving. Numerical results for benchmark problems demonstrate the high accuracy, resolution, efficiency and robustness of the positivity-preserving implicit high-order finite volume method.