No abstract

No abstract

Stable and accurate interface conditions based on the SAT penalty method are derived for the linear advection-diffusion equation. The conditions are functionally independent of the spatial order of accuracy and rely only on the form of the discrete operator. We focus on high-order finite-difference operators that satisfy the summation-by-parts (SBP) property.We prove that stability is a natural consequence of the SBP operators used in conjunction with the new, penalty type, boundary conditions. In addition, we show that the interface treatments are conservative. The issue of the order of accuracy of the interface boundary conditions is clarified. New finite-difference operators of spatial accuracy up to sixth order are constructed which satisfy the SBP property. These finite-difference operators are shown to admit design accuracy ( pth-order global accuracy) when ( p − 1)th-order stencil closures are used near the boundaries, if the physical boundary conditions and interface conditions are implemented to at least pth-order accuracy. Stability and accuracy are demonstrated on the nonlinear Burgers' equation for a 12-subdomain problem with randomly distributed interfaces.

variables needed to evaluate the flux terms. Consequently, the U-vector nmst be decomposed into other variables, leaving the (:-vector itself disposable. Both Williamson (2N) and vdH (2R) schemes may be easily generalized I() accommodate more than two storage registers (N or R). We make no claim that these two strategies are the only viable ones. We do suggest, however, that the vdH methodology is extremely aggressive in its conservation of computer memory usage. In t.he pursui! of computer memory use reduction, the first, casualty is the retention of the U-vector at. the beginning stage. Error control, in the more traditional sense, becomes impossible. A rejected step (such as violation of the error tolerance) cannot be restarted front U (') because with a 2N or 2t1 scheme, U (') is no tongrr available. Instead, alloting one additional register for an error estimate, one may monitor the error nlathematical soft.ware.J18, 28, 29, 58, 88] Attempts were made to solve for schemes symbolically, but il was found that the assumption of various a O = bj quickly made matters intractable because the equations of condition become algebraically nonlinear in the bi's. Scheme coefficients are given to at. lea,st 2,5 digils of accuracy. Some Ell.I{ background is necessary to facilitate a discussion on the optimization of accuracy efficiency, stability efficiency, error control reliability, step-control stability, linear stability, nonlinear stability, and dispersion and dissipation error within the context, of storage reduction in later chapters. This will be done in sections 1 and 2. Two-register schemes will be reviewed in section 3 while three-, four-, and five-register schemes will be considered in sections 4, .5, and 6. Merits of the low-storage schemes are discussed in section P (mininmm phase error), and for "P" melhods, qdisp and qdis_ are the respective dispersion and dissipation orders of accuracy.

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation element methods of arbitrary order for the compressible Navier-Stokes equations. The new methods are similar to strong form, nodal discontinuous Galerkin spectral elements but conserve entropy for the Euler equations and are entropy stable for the Navier-Stokes equations. Shock capturing follows immediately by combining them with a dissipative companion operator via a comparison approach. Smooth and discontinuous test cases are presented that demonstrate their efficacy. Introduction.Next generation numerical algorithms for use in large eddy simulations (LES) and hybrid Reynolds-averaged Navier-Stokes (RANS)-LES simulations will undoubtedly rely on efficient high-order formulations. Although highorder techniques are well suited for LES, most lack robustness when the solution contains discontinuities or even underresolved physical features. Although a variety of stabilization techniques have been developed for second-order methods (e.g., total variation diminishing (TVD) limiters [35], and entropy stability [40]), extending these techniques to high-order formulations has been problematic. High-order essentially nonoscillatory (ENO) [20,36] and weighted ENO (WENO) [31,25] schemes provide a partial remedy to the problem; they achieve high-order design accuracy away from captured discontinuities and maintain sharp "nearly monotone" captured shocks. Unfortunately, nonoscillatory schemes experience instabilities in less than ideal circumstances (e.g., curvilinear mapped grids or expansion of flows into vacuum). Because nonoscillatory schemes are largely based on stencil biasing heuristics rather than stability analysis, there is little theory to guide further development efforts focused on

No abstract

No abstract

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneousapproximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.