•History of SBP operators and SATs, with a focus on the Navier–Stokes (NS) equations.•Methodology for deriving SBP operators for 1st and 2nd derivatives.•Implementation of SBP-SATs for the numerical ...solution of the NS equations.•Review of recent developments, including functional superconvergence.
Summation-by-parts (SBP) operators have a number of properties that make them an attractive option for higher-order spatial discretizations of partial differential equations. In particular, they enable the derivation of higher-order boundary closures leading to provable time stability. When implemented on multi-block structured meshes in conjunction with simultaneous approximation terms (SATs)—penalty terms that impose boundary and interblock-coupling conditions in a weak sense—they offer additional properties of value, even for second-order accurate schemes and steady problems. For example, they involve low communication overhead for efficient parallel algorithms and relax the continuity requirements of both the mesh and the solution across block interfaces. This paper provides a brief history of seminal contributions to, and applications of, SBP-SAT methods followed by a description of their properties and a methodology for deriving SBP operators for first derivatives and second derivatives with variable coefficients. A procedure for deriving SATs is also provided. Practical aspects are discussed, including artificial dissipation, transformation to curvilinear coordinates, and application to the Navier–Stokes equations. Recent developments are reviewed, including a variational interpretation, the connection to quadrature rules, functional superconvergence, error estimates, and dual consistency. Finally, the connection to quadrature rules is exploited to provide a generalization of the SBP concept to a broader class of operators, enabling a unification and rigorous development of SATs for operators such as nodal-based pseudo-spectral and some discontinuous Galerkin operators.
A new formula is derived for the random error of sample central moments from correlated data which does not assume an underlying distribution and is accurate to leading order in the number of sample ...elements. Central moments, being important quantities in turbulence research, require accurate error estimation. Many approaches have been followed in the past for estimating the random errors of central moments from correlated data. These include: simple extensions of the formula for independent data, using the formula for the random error of generic averages, assuming an underlying normal distribution, and using block bootstraps. All of these approaches are compared with the present formula using datasets from a turbulent boundary layer, freestream grid turbulence, and a turbulent round jet. For even-order sample central moments, many of the existing approaches perform well with differences of <15%. However, for odd-order sample central moments, only the block bootstrap methodology performs similarly well. For the same sample central moments, the other methods differ by as much as 200–1000%.
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 h2p when a degree p multidimensional SBP operator is used to discretize the spatial derivatives. SAT penalty coefficients corresponding to various discontinuous Galerkin fluxes developed for elliptic partial differential equations are identified. We demonstrate that the original method of Bassi and Rebay, the modified method of Bassi and Rebay, and the symmetric interior penalty method are equivalent when implemented with SBP diagonal-E operators that have diagonal norm matrix, e.g., the Legendre-Gauss-Lobatto SBP operator in one space dimension. Similarly, the local discontinuous Galerkin and the compact discontinuous Galerkin schemes are equivalent for this family of operators. The analysis remains valid on curvilinear grids if a degree ≤p+1 bijective polynomial mapping from the reference to physical elements is used. Numerical experiments with the two-dimensional Poisson problem support the theoretical results.
•Simultaneous approximation terms for diffusion problems.•Adjoint consistency and functional superconvergence of SBP-SAT discretizations.•Equivalence of simultaneous approximation terms.•SBP-SAT discretization of second-order PDEs with multidimensional SBP operators.•Consistent, conservative, and energy stable SATs for elliptic problems.
In recent decades, the environmental impacts of aviation have become a key challenge for the aeronautical community. Advanced and well-established technologies such as active flow control systems, ...wing-tip devices, high bypass ratio engines, composite materials, among others, have demonstrated fuel-burn benefits by reducing drag and/or weight. Nevertheless, aviation remains under intense pressure to become more sustainable. For this reason, there is a strong drive to explore unconventional aircraft with the aim of reducing both environmental emissions and Direct Operating Cost. This paper presents the current state-of-the-art in the development of future aircraft for civil aviation. The literature review is conducted through an appropriate search protocol to ensure the selection of the most relevant sources. After a brief historical background, progress in the design and development of several unconventional aircraft configurations is presented. Concepts such as Blended/Hybrid Wing Bodies, nonplanar wing designs, next-generation propulsion technologies that are tightly integrated with the airframe, among others, are reviewed. Special attention is given to design methodologies (level-of-fidelity), cruise altitude, aerodynamic performance, and fuel-burn benefits over conventional configurations. The primary contributions of this review are (i) a detailed survey of the design characteristics of unconventional aircraft for non-specialists, and (ii) a comprehensive review of the literature detailing past and current design trends of such configurations for specialists.
Gaussian processes provide probabilistic surrogates for various applications including classification, uncertainty quantification, and optimization. Using a gradient‐enhanced covariance matrix can be ...beneficial since it provides a more accurate surrogate relative to its gradient‐free counterpart. An acute problem for Gaussian processes, particularly those that use gradients, is the ill‐conditioning of their covariance matrices. Several methods have been developed to address this problem for gradient‐enhanced Gaussian processes but they have various drawbacks such as limiting the data that can be used, imposing a minimum distance between evaluation points in the parameter space, or constraining the hyperparameters. In this paper a diagonal preconditioner is applied to the covariance matrix along with a modest nugget to ensure that the condition number of the covariance matrix is bounded, while avoiding the drawbacks listed above. The method can be applied with any twice‐differentiable kernel and when there are noisy function and gradient evaluations. Optimization results for a gradient‐enhanced Bayesian optimizer with the Gaussian kernel are compared with the use of the preconditioning method, a baseline method that constrains the hyperparameters, and a rescaling method that increases the distance between evaluation points. The Bayesian optimizer with the preconditioning method converges the optimality, that is, the ℓ2$$ {\ell}_2 $$ norm of the gradient, an additional 5 to 9 orders of magnitude relative to when the baseline method is used and it does so in fewer iterations than with the rescaling method. The preconditioning method is available in the open source Python library GpGradPy, which can be found at https://github.com/marchildon/gpgradpy/tree/paper_precon.
We present and analyze an entropy-stable semi-discretization of the Euler equations based on high-order summation-by-parts (SBP) operators. In particular, we consider general multidimensional SBP ...elements, building on and generalizing previous work with tensor–product discretizations. In the absence of dissipation, we prove that the semi-discrete scheme conserves entropy; significantly, this proof of nonlinear L2 stability does not rely on integral exactness. Furthermore, interior penalties can be incorporated into the discretization to ensure that the total (mathematical) entropy decreases monotonically, producing an entropy-stable scheme. SBP discretizations with curved elements remain accurate, conservative, and entropy stable provided the mapping Jacobian satisfies the discrete metric invariants; polynomial mappings at most one degree higher than the SBP operators automatically satisfy the metric invariants in two dimensions. In three-dimensions, we describe an elementwise optimization that leads to suitable Jacobians in the case of polynomial mappings. The properties of the semi-discrete scheme are verified and investigated using numerical experiments.
•Entropy-stable, SBP discretizations for non-tensor product, curved elements.•Polynomial mappings can be at most one degree higher than the SBP operators.•3D mapping Jacobian found by solving an elementwise optimization problem.•Numerical experiments verify entropy stability and accuracy.
A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are (i) ...non-repeating interior point operators, (ii) nonuniform nodal distribution in the computational domain, (iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the SBP property. It is proven that the positive-definite norm matrix of each SBP operator must be associated with a quadrature rule; moreover, given a quadrature rule there exists a corresponding SBP operator, where for diagonal-norm SBP operators the weights of the quadrature rule must be positive. The generalized framework gives a straightforward means of posing many known approximations to the first derivative as SBP operators; several are surveyed, such as discontinuous Galerkin discretizations based on the Legendre–Gauss quadrature points, and shown to be SBP operators. Moreover, the new framework provides a method for constructing SBP operators by starting from quadrature rules; this is illustrated by constructing novel SBP operators from known quadrature rules. To demonstrate the utility of the generalization, the Legendre–Gauss and Legendre–Gauss–Radau quadrature points are used to construct SBP operators that do not include one or both boundary nodes.
We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. ...Within such an algebraic framework, the multidimensional summation-by-parts (SBP) property is used to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties of a broad class of DG and FR schemes. Specifically, the analysis enables the extension of the equivalence between the strong and weak forms of the discontinuous Galerkin collocation spectral-element method demonstrated by Kopriva and Gassner (J Sci Comput 44:136–155, 2010) to more general nodal and modal DG formulations, as well as to the Vincent-Castonguay-Jameson-Huynh (VCJH) family of FR methods. Moreover, new algebraic proofs of conservation and energy stability for DG and VCJH schemes with respect to suitable quadrature rules and discrete norms are presented in which the SBP property serves as a unifying mechanism for establishing such results. Numerical experiments are provided for the two-dimensional linear advection and Euler equations, highlighting the design choices afforded for methods within the proposed framework and corroborating the theoretical analysis.
We analyze the stability and functional superconvergence of discretizations of diffusion problems with the narrow-stencil second-derivative generalized summation-by-parts (SBP) operators coupled with ...simultaneous approximation terms (SATs). Provided that the primal and adjoint solutions are sufficiently smooth and the SBP-SAT discretization is primal and adjoint consistent, we show that linear functionals associated with the steady diffusion problem superconverge at a rate of 2
p
when a degree
p
+
1
narrow-stencil or a degree
p
wide-stencil generalized SBP operator is used for the spatial discretization. Sufficient conditions for stability of adjoint consistent discretizations with the narrow-stencil generalized SBP operators are presented. The stability analysis assumes nullspace consistency of the second-derivative operator and the invertibility of the matrix approximating the first derivative at the element boundaries. The theoretical results are verified by numerical experiments with the one-dimensional Poisson problem.
This article presents constrained numerical optimization of high-order linearly and algebraically stable diagonally-implicit Runge–Kutta methods. After satisfying the desired order conditions, ...undetermined coefficients are optimized with respect to objective functions which consider accuracy, stability, and computational cost. Constraints are applied during the optimization to enforce stability properties, to ensure a well-conditioned method, and to limit the domain of the abscissa. Two promising third-order methods are derived using this approach, labelled SDIRK3,(1,2,2)(3)L_14 and SDIRK3,1(4)L_SA_5. Both optimized schemes have a good balance of properties. The relative error norm of the latter, the L2-norm scaled by a function of the number of implicit stages, is a factor of two smaller than comparable methods found in the literature. Variations on these methods are discussed relative to trade-offs in their accuracy and stability properties. A novel fifth-order scheme SDIRK5,1(5)L_02 is derived with a significantly lower relative error norm than the comparable fifth-order A-stable reference method. In addition, the optimized scheme is L-stable. The accuracy and relative efficiency of the Runge–Kutta methods are verified through numerical simulation of van der Pol's equation, as well as numerical simulation of vortex shedding in the laminar wake of a circular cylinder, and in the turbulent wake of a NACA 0012 airfoil. These results demonstrate the value of numerical optimization for selecting undetermined coefficients in the construction of high-order Runge–Kutta methods with a balance between competing objectives.
•Presents a numerical tool for the constrained optimization of Runge–Kutta methods.•Applied to high-order unconditionally stable diagonally-implicit Runge–Kutta methods.•Novel methods with up to a factor of two improvement in predicted relative efficiency.•Theoretical results verified by numerical experiments.
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