Finite element and implicit Runge-Kutta implementation of an acoustics-convection upstream resolution algorithm for the time-dependent two-dimensional Euler equations

摘要

The second of a two-paper series, this paper details a solver for the characterisfics-bias system from the acoustics-convection upstream resolution algorithm for the Euler and Navier-Stokes equations. An integral formulation leads to several Surface integrals that allow effective enforcement of boundary conditions. Also presented is a new multi-dimensional procedure to enforce a pressure boundary condition at a Subsonic outlet, a procedure that remains accurate and stable. A classical finite element Galerkin discretization of the integral formulation on any prescribed grid directly yields an optimal discretely conservative upstream approximation for the Euler and Navier-Stokes equations, an approximation that remains multi-dimensional independently of the orientation of the reference axes and computational cells. The time-dependent discrete equations are then integrated in time via an implicit Runge-Kutta procedure that in this paper is proven to remain absolutely non-linearly stable for the spatially-discrete Euler and Navier-Stokes equations and shown to converge rapidly to steady states, with maximum Courant number exceeding 100 for the linearized version. Even on relatively coarse grids, the acoustics-convection Upstream resolution algorithm generates essentially non-oscillatory Solutions for Subsonic, transonic and supersonic flows, encompassing oblique- and interacting-shock fields that converge within 40 time steps and reflect reference exact Solutions. Copyright (c) 2005 John Wiley & Sons, Ltd.