Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area

Entropy solutions of the compressible Euler equations

We consider the three-dimensional Euler equations of gas dynamics on a bounded periodic domain and a bounded time interval. We prove that Lax-Friedrichs scheme can be used to produce a sequence of solutions with ever finer resolution for any appropriately bounded (but not necessarily small) initial data. Furthermore, we show that if the density remains strictly positive in the sequence of solutions at hand, a subsequence converges to an entropy solution. We provide numerical evidence for these results by computing a sensitive Kelvin-Helmholtz problem

Large Eddy Simulations by approximate weak entropy solutions

We propose to use weak (averaged) entropy solutions in lieu of LES models. This approach unites the theory for shock capturing schemes and turbulence modelling. To achieve this, we identify a number of conditions (albeit not sufficient) that a scheme should satisfy. Namely, a scheme should be: conservative, entropy dissipative, kinetic-energy preserving/diffusive, positivity preserving and have linearly stable (non anti-diffusive) continuity equation. We propose a finite-volume scheme with these properties and investigate its properties, and the properties of some related schemes, for the standard entropy-wave/shock interaction, a Kelvin-Helmoholtz instability, and a turbulent Rayleigh-Taylor problem. These preliminary investigations suggest that the scheme is very robust, is competitive for turbulent problems and is far less prone to trip false turbulence. However, and as with any general purpose scheme, wildly under-resolved simulations can not be expected to be accurate. The advantage with the current scheme is that local averages converge which provides a possibility to estimate the accuracy of functionals of interest.publishedVersio

Entropy stable boundary conditions for the Euler equations

We consider the initial-boundary value Euler equations with the aim to derive boundary conditions that yield an entropy bound for the physical (Navier-Stokes) entropy. We begin by reviewing the entropy bound obtained for standard no-penetration wall boundary conditions and propose a numerical implementation. The main results are the derivation of full-state boundary conditions (far-field, inlet, outlet) and the accompanying entropy stable implementations. We also show that boundary conditions obtained from linear theory are unable to bound the entropy and that non-linear bounds require additional boundary conditions. We corroborate our theoretical findings with numerical experiments.publishedVersio

Convergence of Chandrashekar’s Second-Derivative Finite-Volume Approximation

We consider a slightly modified local finite-volume approximation of the Laplacian operator originally proposed by Chandrashekar (Int J Adv Eng Sci Appl Math 8(3):174–193, 2016, https://doi.org/10.1007/s12572-015-0160-z). The goal is to prove consistency and convergence of the approximation on unstructured grids. Consequently, we propose a semi-discrete scheme for the heat equation augmented with Dirichlet, Neumann and Robin boundary conditions. By deriving a priori estimates for the numerical solution, we prove that it converges weakly, and subsequently strongly, to a weak solution of the original problem. A numerical simulation demonstrates that the scheme converges with a second-order rate.publishedVersio

A study of the diffusive properties of a modified compressible Navier-Stokes model

The aim of this study is to provide further validation for the weakly well-posed modified compressible Navier-Stokes system proposed in Svärd (Phys A 506:350–375, 2018) when applied to ideal gases. We do so by considering sound attenuation, both theoretically and numerically for argon and oxygen, and make comparisons with experimental values in the literature. Furthermore, we compute shock profiles for argon and nitrogen, and compare them with experiments in the literature. Our numerical simulations have revealed problems when using experimental attenuation data, as presented in the available literature, for validation and determination of diffusion coefficients. However, comparisons with the shock data, suggest that the modified system may benefit from an additional heat diffusive term. In view of these and previously published validation tests, the model proposed in Svärd (Phys A 506:350–375, 2018) is equally accurate as the standard compressible system. However, with more complete experimental information for the attenuation case at hand, it might be possible to further improve the accuracy by more precise determination of the diffusion coefficients. We propose a tentative adjustment of the model that may be tested/validated, if more detailed experimental information becomes available.publishedVersio

Entropy stability for the compressible Navier-Stokes equations with strong imposition of the no-slip boundary condition

We consider the compressible Navier-Stokes equations subject to no-slip adiabatic wall boundary conditions. The main goal is to investigate stability properties of schemes imposing the no-slip condition strongly (injection) and the temperature condition weakly by a simultaneous approximation term. To this end, we propose a low-order summation-by-parts scheme. By verifying the complete linearisation procedure, we prove linear stability for the scheme. In addition, and assuming that the interior scheme is entropy stable, we also prove entropy stability for the full scheme including the boundary treatment. Furthermore, we propose a linearly stable 3rd-order scheme with the same imposition of the wall conditions. However, the 3rd-order scheme is not provably non-linearly stable. A number of simulations show that the boundary procedure is robust for both schemes.publishedVersio

Stability issues of entropy-stable and/or split-form high-order schemes -- Analysis of local linear stability

The focus of the present research is on the analysis of local linear stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local linear stability is not guaranteed even when the scheme is non-linearly stable and that this has grave implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally linearly unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we demonstrate numerically that such schemes cause exponential growth of errors. Furthermore, we demonstrate a similar abnormal behaviour for the compressible Euler equations. Finally, we demonstrate numerically that other commonly used split-forms, such as the Kennedy and Gruber splitting, are also locally linearly unstable

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.Comment: 20 page