Summation-By-Parts (SBP) methods provide a systematic way of constructing provably stable numerical schemes. However, many proofs of convergence and accuracy rely on the assumption that the SBP operator possesses a particular eigenvalue property. In this note, three results pertaining to this property are proven. Firstly, the eigenvalue property does not hold for all nullspace consistent SBP operators. Secondly, this issue can be addressed without affecting the accuracy of the method by adding a specially designed, arbitrarily small perturbation term to the SBP operator. Thirdly, all pseudospectral methods satisfy the eigenvalue property.
A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
ETSIAE-UPM - School of Aeronautics, Universidad Politécnica de Madrid（马德里理工大学航空学院）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Computational Science Research Center, San Diego State University（圣地亚哥州立大学计算科学研究中心）
We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressiblity, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split-form nodal Discontinuous Galerkin (DG) approximation…
Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of $A$ stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the in…
We consider finite difference approximations of the second derivative, exemplified in Poisson's equation, the heat equation and the wave equation. The finite difference operators satisfy a summation-by-parts property, which mimics the integration-by-parts. Since the operators approximate the second derivative, they are singular by construction. To impose boundary conditions, these operators are modified using Simultaneous Approximation Terms. This makes the modified matrices non-singular, for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, when c…
An Entropy Stable h/p Non-Conforming Discontinuous Galerkin Method with the Summation-by-Parts Property
Mathematical Institute, University of Cologne（科隆大学数学研究所）；National Institute of Aerospace and Computational AeroSciences Branch, NASA Langley Research Center（NASA兰利研究中心国家航空航天与计算空气动力学分支）；Computational AeroSciences Branch, NASA Langley Research Center（NASA兰利研究中心计算空气动力学分支）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
We develop an entropy stable two-phase incompressible Navier--Stokes/Cahn--Hilliard discontinuous Galerkin (DG) flow solver method. The model poses the Cahn-Hilliard equation as the phase field method, a skew-symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure. We design the model so that it satisfies an entropy law, including free- and no-slip wall boundary conditions with non-zero wall contact angle. We then construct a high-order DG approximation of the model that satisfies the SBP-SAT property. With the help of a discrete stability anal…
The entropy conservative, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernandez et al. (2019), is extended from the compressible Euler equations to the compressible Navier-Stokes equations. A simple and flexible coupling procedure with planar interpolation operators between adjoining nonconforming elements is used. Curvilinear volume metric terms are numerically approximated via a minimization procedure and satisfy the discrete geometric conservation law conditions. Distinct curvilinear surface metrics are used on the adjoining interfaces to c…
Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws
University of Cologne（科隆大学）；Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Center for Data and Simulation Science (CDS), University of Cologne（科隆大学数据与模拟科学中心）
In this paper, the entropy conservative/stable algorithms presented by Del Rey Fernandez and coauthors [18,16,17] for the compressible Euler and Navier-Stokes equations on nonconforming p-refined/coarsened curvilinear grids is extended to h/p refinement/coarsening. The main difficulty in developing nonconforming algorithms is the construction of appropriate coupling procedures across nonconforming interfaces. Here, a computationally simple and efficient approach based upon using decoupled interpolation operators is utilized. The resulting scheme is entropy conservative/stable and element-wise …
We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material…
The entropy conservative/stable algorithm of Friedrich~\etal (2018) for hyperbolic conservation laws on nonconforming p-refined/coarsened Cartesian grids, is extended to curvilinear grids for the compressible Euler equations. The primary focus is on constructing appropriate coupling procedures across the curvilinear nonconforming interfaces. A simple and flexible approach is proposed that uses interpolation operators from one element to the other. On the element faces, the analytic metrics are used to construct coupling terms, while metric terms in the volume are approximated to satisfy a disc…
Entropy Stable Space-Time Discontinuous Galerkin Schemes with Summation-by-Parts Property for Hyperbolic Conservation Laws
University of Cologne（科隆大学）；National Institute of Aerospace and Computational AeroSciences Branch, NASA Langley Research Center（NASA兰利研究中心国家航空航天与计算空气动力学分支）；Computational AeroSciences Branch, NASA Langley Research Center（NASA兰利研究中心计算空气动力学分支）
Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix--Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Ravia…
High-Order Summation-By-Parts Schemes for First-Order Hyperbolic Systems in Curvilinear Coordinates with Singularities
Radboud University（拉德堡德大学）；Albert-Einstein-Institut, Max-Planck-Institut für Gravitationsphysik（马克斯·普朗克引力物理学研究所爱因斯坦研究所）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）；Louisiana State University（路易斯安那州立大学）
3d Summation-by-Parts scheme for Linear Wave Equations on Hyperboloidal Slices
Indian Institute of Science Education and Research (IISER) Pune（印度科学教育研究所浦那分校）；International Centre for Theoretical Sciences (ICTS), Tata Institute of Fundamental Research (TIFR)（塔塔基础研究所国际理论科学中心）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学科学与应用研究院）
