Design and analysis of robust ALE time-integrators for the solution of unsteady flow problems on moving grids

Abstract

Two methodologies for designing an arbitrary Lagrangian–Eulerian (ALE) time-integrator for the semi-discrete Navier–Stokes equations are reviewed. Each methodology consists of a different mathematical framework for extending to moving grids a numerical scheme originally developed for computational fluid dynamics (CFD) on fixed grids, while preserving its formal order of time-accuracy established on fixed grids. Given a favorite scheme for the solution on fixed grids of the discrete Navier–Stokes equations, each of these two mathematical frameworks can generate multiple ALE extensions that share the same order of time-accuracy on moving grids. Typically, only a subset of these ALE schemes satisfy their respective discrete geometric conservation laws (DGCLs). Next, using a nonlinear scalar conservation law (NSCL) as a model problem, it is proved that satisfying the corresponding DGCL is a necessary condition for any ALE scheme to preserve on moving grids the nonlinear stability properties of its fixed-grid counterpart. Using the same NSCL, it is also proved that for the ALE extension of the second-order time-accurate trapezoidal rule, the DGCL requirement is a necessary as well as sufficient condition for nonlinear stability on moving grids. Hence, each of the two mathematical frameworks described in this paper combined with the DGCL test provides a general methodology for designing a robust ALE time-integrator for the solution of unsteady flow problems on dynamic meshes. As an example, the ALE extension of the popular three-point backward difference scheme is discussed in details. The corresponding theoretical results are illustrated with its application to the solution of various unsteady flow problems arising from the vibrations of the AGARD Wing 445.6 as well as a complete F-16 configuration in transonic airstreams.