Temporally adaptive conservative scheme for unsteady compressible flow

Abstract

An adaptive conservative time integration scheme (ACTI) for compressible flow simulations is devised. A finite volume method is used, in which every cell employs a time step size which is some power of two times smaller than a global time step. By first advancing those cells with smaller time steps, this allows to use the already computed accumulated fluxes when the adjacent cells with larger time steps catch up. Conservation and periodic synchronization are thus guaranteed by construction; both important advantages of the new scheme (especially in the presence of shock waves) compared to previously proposed adaptive time stepping methods for compressible flow. Accuracy and computational speedup are demonstrated for challenging 1D and 2D test cases. Comparisons of ACTI results with an analytical solution and with results obtained without ACTI show excellent agreement. Although maximum wave speeds and/or cell sizes vary by several orders of magnitude throughout the computational domains, the CFL numbers employed by ACTI in each cell vary not more than by approximately a factor of two. Without ACTI, on the other hand, the CFL numbers are tiny in most of the cells. Since this results in a dramatic reduction of required flux computations, high speedup gains are achieved due to ACTI; in particular, if only few cells experience a very small time step, while the time step size limitations are less severe for the majority of cells.