ASSESSMENT OF LOCAL PRECONDITIONERS FOR STEADY STATE AND TIME DEPENDENT FLOWS Veer N. Vatsa * NASA Langley Research Center, Hampton,VA Eli Turkel † Tel-Aviv University, Israel and NIA, Hampton, VA Abstract Preconditioners for hyperbolic systems are numerical artifices intended to accelerate the convergence path to a steady state. In addition, in some cases, the preconditioner can also be included in the artificial viscosity/upwinding terms in order to improve the accuracy of the steady state solution. For time dependent problems we use a dual time stepping approach; and therefore the preconditioner affects the convergence rate and accuracy of the subiterations at each physical time step. We consider two types of local preconditioners that couple the governing equations at a node point: Jacobi and low-speed preconditioning. We consider their effectiveness for both steady state and time dependent problems with regard to the convergence rate and the numerical accuracy. We also consider the effect of the far field boundary conditions on both steady state and time dependent problems. 1 Introduction Preconditioning methods for low-speed, steady flows have been available for almost twenty years [16]. Because such precon- ditioners are generally designed to modify the path to steady state, special attention is required for adapting these methods for unsteady flows to maintain temporal accuracy. Fortunately, such preconditioning techniques are easily extendable to unsteady flow applications in conjunction with dual time stepping algorithms. We consider the hyperbolic system of unsteady Euler equations appended with pseudo time derivatives wτ + ξwt + Awx + Bwy + Cwz =0 (1) where the flux Jacobian matrices A,B,C are symmetric (or simultaneously symmetrizable). We are interested in τ →∞, where τ is viewed as an artificial (pseudo) time. For physically steady state problems ξ =0, while for time dependent problems ξ is set equal to 1. We discretize the physical time derivative with a backward difference formula (BDF) ∂w ∂t ≃ ct Δt w n+1 + Q(w n ,w n−1 , ...) Δt (2) where, ct is a constant that depends on the order of the BDF scheme. Define ζ = ξc t Δt . We Fourier transform Eq. (1) in space and replace wt by Eq. (2). Define the amplification matrix G(ω1,ω2,ω3)= −ζI + i(Aω1 + Bω2 + Cω3) (3) The condition number of this system is defined as cond# = max ω i     λmax(G) λmin(G)     . (4) where λ denotes an eigenvalue of the matrix while ωi are real numbers with ω 2 1 + ω 2 2 + ω 2 3 =1. Note that the eigenvalues of i(Aω1 + Bω2 + Cω3) are pure imaginary since the matrices are symmetric. Physically the condition number (with ξ =0) can ∗ Senior Member † Professor, Department of Mathematics, Associate Fellow 1