A scalable fully implicit method with adaptive time stepping for unsteady compressible inviscid flows

Abstract

The class of fully implicit methods is drawing more attention in the simulation of fluid dynamics for engineering community, due to the allowance of large time steps in extreme-scale simulations. In this paper, we introduce and study a scalable fully implicit method for the numerical simulations of unsteady compressible inviscid flows governed by the compressible Euler equations. In the method, a cell-centered finite volume scheme together with the local Lax-Friedrichs (LLF) formula is used for the spatial discretization, and a backward differentiation formula is applied to integrate the Euler equations in time. The resultant nonlinear system at each time step is then solved by a parallel Newton-Krylov method with a domain decomposition type preconditioner. To improve the performance of the proposed method, we introduce an adaptive time stepping method which adjusts the time step size according to the initial residual of Newton iterations. Therefore, the proposed fully implicit solver overcomes the often severe limits on the time steps associated with existing methods. Numerical experiments validate that the approach is effective and robust for the simulations of several compressible inviscid flows. We also show that the newly developed algorithm scales well with more than one thousand processor cores for the problem with tens of millions of unknowns.