A Scalable Fully Implicit Compressible Euler Solver for Mesoscale Nonhydrostatic Simulation of Atmospheric Flows

Abstract

A fully implicit solver is developed for the mesoscale nonhydrostatic simulation of atmospheric flows governed by the compressible Euler equations. To spatially discretize the Euler equations on a height-based terrain-following mesh, we apply a cell-centered finite volume scheme, in which an advection upstream splitting (AUSM$^{+}$-up) method with a piecewise linear reconstruction is employed to achieve second-order accuracy for the low-Mach flow. A second-order explicit-first-step, single-diagonal-coefficient, diagonally implicit Runge--Kutta (ESDIRK) method with adaptive time stepping is applied to stabilize physically insignificant fast waves and accurately integrate the Euler equations in time. The nonlinear system arising at each time step is solved by using a Jacobian-free Newton--Krylov--Schwarz (NKS) algorithm. To accelerate the convergence and improve the robustness, we employ a class of additive Schwarz preconditioners in which the subdomain Jacobian matrix is constructed using a first-order spatial discretization. Several test cases are used to validate the correctness of the scheme and examine the performance of the solver. Large-scale results on a supercomputer with up to 18,432 processor cores are provided to show the parallel performance of the proposed method.