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 initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.
Entropy stable schemes replicate an entropy inequality at the semi-discrete level. These schemes rely on an algebraic summation-by-parts (SBP) structure and a technique referred to as flux differencing. We provide simple and efficient formulas for Jacobian matrices for the semi-discrete systems of ODEs produced by entropy stable discretizations. These formulas are derived based on the structure of flux differencing and derivatives of flux functions, which can be computed using automatic differentiation (AD). Numerical results demonstrate the efficiency and utility of these Jacobian formulas, w…
Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop and construct energy-stable radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite …
The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble global SBP operators and use continuous solutions, unlike previous efforts that use discontinuous SBP discretizations. Derivative-based dissipation and local-projection stabilization (LPS) are investigated as options for stabilizing the baseline discretization. These stabilizations are equal up to a multiplicative constant in one dimension, but only LPS remains well conditioned for general, multidimensional SBP operators.…
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…
A finite-difference summation-by-parts, conditionally stable partitioned algorithm for conjugate heat transfer problems
University of Waterloo（滑铁卢大学）；Ansys Canada Inc.（安世加拿大公司）；the University of British Columbia（不列颠哥伦比亚大学）；Institute of Mechanical and Electrical Engineering, University of Southern Denmark（南丹麦大学机电工程研究所）
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）
Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show that these summation formulas actually arise as particular instances of a single series expansion, including Euler's method for alternating series. This new summation formula gives rise to a family of polynomials, which contain both the Bernoulli and Gregory numbers in their coefficients. We prove some properties of those polynomials, such as recurrence ide…
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…
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 pseudospe…
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）
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 …
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…
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…
A high order accurate and energy stable continuous Galerkin framework on summation-by-parts form for the incompressible Navier-Stokes equations
InCFD Research Group, Department of Mechanical Engineering, University of Cape Town（开普敦大学机械工程系）；Department of Mathematical Sciences, Applied Mathematics Division, Stellenbosch University（斯泰伦博斯大学应用数学系）；Department of Mathematics, Applied Mathematics, Linköping University（林雪平大学应用数学系）；Department of Mathematics and Applied Mathematics, University of Johannesburg（约翰内斯堡大学应用数学系）
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）
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）
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…
The construction of high order entropy stable collocation schemes on quadrilateral and hexahedral elements has relied on the use of Gauss-Legendre-Lobatto collocation points and their equivalence with summation-by-parts (SBP) finite difference operators. In this work, we show how to efficiently generalize the construction of semi-discretely entropy stable schemes on tensor product elements to Gauss points and generalized SBP operators. Numerical experiments suggest that the use of Gauss points significantly improves accuracy on curved meshes.
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to "upwind" the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a …
The focus of the present research is on the analysis of local energy 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 energy stability, i.e., the numerical growth rate does not exceed the growth rate of the continuous problem, is not guaranteed even when the scheme is non-linearly stable and that this may have adverse implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally energy unstable, as the…
Several types of simultaneous approximation term (SAT) for diffusion problems discretized with diagonal-norm multidimensional summation-by-parts (SBP) operators are analyzed based on a common framework. Conditions under which the SBP-SAT discretizations are consistent, conservative, adjoint consistent, and energy stable are presented. For SATs leading to primal and adjoint consistent discretizations, the error in output functionals is shown to be of order $h^{2p}$ when a degree $p$ multidimensional SBP operator is used to discretize the spatial derivatives. SAT penalty coefficients correspondi…
We present an extended range of stable flux reconstruction (FR) methods on triangles through the development and application of the summation-by-parts framework in two-dimensions. This extended range of stable schemes is then shown to contain the single parameter schemes of \citet{Castonguay2011} on triangles, and our definition enables wider stability bounds to be developed for those single parameter families. Stable upwinded spectral difference (SD) schemes on triangular elements have previously been found using Fourier analysis. We used our extended range of FR schemes to investigate the li…
Strongly stable dual-pairing summation by parts finite difference schemes for the vector invariant nonlinear shallow water equations -- I: Numerical scheme and validation on the plane
