Parallel Newton-Krylov Solver for the Euler equations Discretized Using Simultaneous Approximation Terms

Abstract

We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations on multiblock structured meshes. The Euler equations are discretized on each block independently using second-order-accurate summation-by-parts operators and scalar numerical dissipation. Boundary conditions are imposed and block interfaces are coupled using simultaneous-approximation terms. The summation-by-parts with simultaneousapproximation-terms approach is time-stable, requires only C 0 mesh continuity at block interfaces, accommodates arbitrary block topologies, and has low interblock-communication overhead. The resulting discrete equations are solved iteratively using an inexact-Newton method. At each Newton iteration, the linear system is solved inexactly using a Krylov-subspace iterative method, and both additive Schwarz and approximate Schur preconditioners are investigated. The algorithm is tested on the ONERA M6 wing geometry. We conclude that the approximate Schur preconditioner is an efficient alternative to the Schwarz preconditioner. Overall, the results demonstrate that the Newton-Krylov algorithm is very efficient: using 24 processors, a transonic flow on a 96-block, 1-million-node mesh requires 12 minutes for a 10-order reduction of the residual norm.

Optimal diagonal-norm SBP operators

“…To show the importance of using diagonal-norm SBP operators for this problem we present in Figure 4 the eigenvalues to (17), on a grid defined by (14) with l = 2 and m = 51. We compare the spectra using the optimal 8th order diagonal-norm SBP operator and the corresponding block-norm The accuracy properties will be tested for an analytic standing wave solution, across a discontinuous media interface (located at ξ = 0),…”

“…The SBP-SAT method combines semi-discrete operators that satisfy a summation-by-parts (SBP) formula [20], with physical boundary conditions implemented using the simultaneous approximation term (SAT) method [5]. Examples of the SBP-SAT approach can be found in [33,34,35,28,30,31,36,27,43,23,9,29,17,16,19].…”

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

“…Lerat and Wu [18] adopted local flux construction to establish conservative and unconditionally stable interface conditions. Huan, Hicken and Zingg [19,20] proposed a kind of high-order interface boundary schemes which combine a conventional scheme and a summation-by-parts (SBP) [21][22][23] scheme with simultaneous approximation terms (SATs) [24][25][26]. The SBP operators are derived from the energy method, first by Kreiss and Scherer in 1974 [21] for low orders, and extended to high-order by Strand [27] and Jurgens [28].…”

A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain

“…The SAT technique (weak imposition of boundary condition or penalty technique) is normally applied only at one grid point (as in the example above) [2,3,22,23,31,36,37,44,46]. However, in some cases, it is possible to impose the weak boundary conditions at multiple grid points.…”