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

Abstract

SUMMARYThe second of a two-paper series, this paper details a solver for the characteristics-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 e ective 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 ÿnite 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 ows, encompassing oblique-and interacting-shock ÿelds that converge within 40 time steps and re ect reference exact solutions.

Derivation and characteristics analysis of an acoustics-convection upstream resolution algorithm for the two-dimensional Euler and Navier-Stokes equations

“…For threedimensional formulations, 16j63, and with R denoting the real-number ÿeld, the independent variable (x; t), x ≡ (x 1 ; x 2 ; x 3 ), in (1) When the 'viscosity' ux f j , 16j63, identically vanishes, this system is hyperbolic when the eigenvalues of the Jacobian matrix (@f j (q)=@q)n j are all real for arbitrary unit vectors n with direction cosines n j , the components of n along the coordinate axes. The system is also termed 'strongly' hyperbolic when this matrix possess a full set of eigenvectors [21,22].…”

“…The integral formulation for system (1), [23,24], seeks a solution q ∈ H 1 ( ), subject to prescribed boundary conditions on @ ≡ \ , such that for all test functionsŵ ∈ H where n j , 16j63, denotes the jth component of the outward pointing unit vector perpendicular to @ . For 2-D ows, 16j62, the dependent-variable array q = q(x; t) as well as the inviscid and viscous ux 'vector' components f j and f j are then deÿned as …”

“…With respect to an inertial Cartesian reference frame, with implied summation on repeated indices, the classical Navier-Stokes and Euler conservation law system [19,20] is @q @t + @f j (q) @x j − @f j @x j = 0 (1) which consists of the continuity, linear-momentum and total-energy equations. For threedimensional formulations, 16j63, and with R denoting the real-number ÿeld, the independent variable (x; t), x ≡ (x 1 ; x 2 ; x 3 ), in (1) When the 'viscosity' ux f j , 16j63, identically vanishes, this system is hyperbolic when the eigenvalues of the Jacobian matrix (@f j (q)=@q)n j are all real for arbitrary unit vectors n with direction cosines n j , the components of n along the coordinate axes.…”

“…non-discrete, multi-dimensional and inÿnite-directional characteristics-bias approximations of the Euler and Navier-Stokes equations and subsequent computational implementation. The second part [1], also featured in this Journal, presents a ÿnite element and implicit Runge-Kutta implementation and applications to smooth and shocked aerodynamic and gas dynamic transonic and supersonic ows. This paper details the development and characteristics analysis of the Acoustic-Convection Upstream Resolution algorithm.…”

“…This paper details the development and characteristics analysis of the Acoustic-Convection Upstream Resolution algorithm. 1235 approximation of this companion system then automatically and directly generates a genuinely multi-dimensional upstream-bias discrete analogue of the given system [1].…”