Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

Abstract

This paper introduces alternating-direction implicit (ADI) solvers of higher order of timeaccuracy (orders two to six) for the compressible Navier-Stokes equations in two-and threedimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas-Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDFbased ADI algorithms proposed in this paper are "quasi-unconditionally stable" in the following sense: each algorithm is stable for all couples (h, ∆t) of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form (0, M h ) × (0, M t ). In other words, for each fixed value of ∆t below a certain threshold, the Navier-Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier-Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier-Stokes solvers for which second order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions.

On Multistep Stabilizing Correction Splitting Methods with Applications to the Heston Model

“…To overcome this, we will follow an idea of Bruno & Cubillos [5] who studied BDF splitting methods for linearized Navier-Stokes equations using two different extrapolation formulas in the prediction stage: high-order extrapolation for the explicit term F 0 and a formula of one order lower for the implicit terms F 1 , . .…”

High-Performance Computational Electromagnetics in Frequency-Domain and Time-Domain

“…Significant progress occurred over the life of this contract in the area of FC (Fourier Continuation) methods for Partial Differential Equations in the time-domain [1][2][3][4][5][6][7]. Our efforts in these areas have resulted in explicit and implicit solvers for high-and low-frequency problems, for linear and nonlinear equations, and including media such as fluids, solids and vacuum-and combinations thereof.…”

“…In each of the aforementioned publications a significant milestone was achieved. For example, the contributions [1,7] provide methods that can be used to enable FC solution of nonlinear equations (such as the Burgers and Navier-Stokes equations) while maintaining high-order accuracy and dispersionlessness with quasi-unconditional stability: arbitrarily small values of ∆x can be used for a fixed ∆t, provided the ∆t value adequately samples the problem. In the contribution [2], in turn, methods for FC solution of problems containing variable coefficients were introduced; in particular it was found that certain numerical boundary layers need to be adequately represented in order to ensure accurate solution.…”

Higher-order implicit-explicit multi-domain compressible Navier-Stokes solvers

“…Later, various authors [7,8] demonstrated solvers based on high-order compact finite differences and explicit fourth-order Runge-Kutta time marching as well as an implicit Beam and Warming scheme [9,10] of nominal second-order accuracy (cf. [11]). More recently, the FC methodology was combined with an overset grid approach for the solution of the compressible Navier-Stokes equations in two dimensions [12] and the elasticity equations in three dimensions [13].…”

“…However, in practice, previous work in the context of the Beam and Warming method had not demonstrated temporal accuracies beyond first order (cf. [11]). Nevertheless, high-order time accuracy is crucial in long-time simulations or highly-inhomogeneous flows-for which the dispersion inherent in low-order approaches would make it necessary to use inordinately small time-steps.…”