A Parallel Newton-Krylov-Schur Flow Solver for the Navier-Stokes Equations Using the SBP-SAT Approach

Abstract

This paper presents a three-dimensional Newton-Krylov flow solver for the NavierStokes equations which uses summation-by-parts (SBP) operators on multi-block structured grids. Simultaneous approximation terms (SAT's) are used to enforce the boundary conditions and the coupling of block interfaces. The discrete equations are solved iteratively with an inexact Newton method. The linear system of each Newton iteration is solved using a Krylov subspace iterative method with an approximate-Schur parallel preconditioner. The algorithm is validated against an established two-dimensional flow solver. Additionally, results are presented for laminar flow around the ONERA M6 wing, as well as low Reynolds number flow around a sphere. Using 384 processors, the solver is capable of obtaining the steady-state solution (reducing the flow residual by 12 orders of magnitude) on a 4.1 million node grid around the ONERA M6 wing in 4.2 minutes. Convergence to 3 significant figures in force coefficients is achieved in 83 seconds. Parallel scaling tests show that the algorithm scales well with the number of processors used. The results show that the SBP-SAT discretization, solved with the parallel Newton-Krylov-Schur algorithm, is an efficient option for three-dimensional Navier-Stokes solutions, with the SAT's providing several advantages in enforcing boundary conditions and block coupling.

Drag Minimization Based on the Navier–Stokes Equations Using a Newton–Krylov Approach

“…Boundary conditions and inter-block coupling are enforced weakly using simultaneous approximation terms (SATs) [49][50][51][52]; this approach requires only boundary information from neighboring blocks, minimizing computational overhead. Further details of the SBP-SAT implementation can be found in [46,47,[53][54][55][56][57][58],…”

Superconvergent Functional Estimates from Summation-By-Parts Finite-Difference Discretizations

“…Subsequently, the SBP-SAT method was extended to handle block interfaces [5], curvilinear domains [24], and nonlinear problems requiring dissipation [22]. The SBP-SAT combination has proven to be a powerful approach, with applications including the Euler [22,12], Navier-Stokes [23,25,26], and Einstein equations [17,27].Svärd [32] showed that diagonal-norm SBP operators are necessary to guarantee time stability when coordinate transformations are used. Unfortunately, to achieve both stability and high-order accuracy, diagonal-norm SBP operators have interior stencils that are twice the accuracy of their boundary stencils.…”

Weak and strong wall boundary procedures and convergence to steady-state of the Navier–Stokes equations

“…The main reason to use weak boundary procedures stems from the fact that together with summation-by-parts operators they lead to provable stable schemes. For application of this technique to finite difference methods, node-centered finite volume methods, spectral domain methods and various hybrid methods see [25,42,4,30,31,36,44,48,20,39,6,22,13,24], [32,47,45,46,14,41], [18,16,19,7] and [33,34,15,37,5] respectively. In this paper we will consider a new effect of using weak boundary procedures, namely that it in many cases (all that we tried) speeds up the convergence to steady-state.…”