Table 1 - uploaded by David C. Del Rey Fernández
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The comprehensive generalization of summation-by-parts of Del Rey Fern\'andez
et al.\ (J. Comput. Phys., 266, 2014) is extended to approximations of second
derivatives with variable coefficients. This enables the construction of
second-derivative operators with one or more of the following characteristics:
i) non-repeating interior stencil, ii) non...
Citations
... Examples of schemes based on SBP operators include finite difference (FD) [40,41,70,73], essentially non-oscillatory (ENO) and weighted ENO (WENO) [80,14,5], continuous Galerkin (CG) [1,2], discontinuous Galerkin (DG) [19,7], finite volume (FV) [55,56], flux reconstruction (FR) [36,68,63], and implicit time integration [60,44,67] methods. Second-derivative SBP operators have been the subject of significant research attention as well [50,47,12], for instance in the context of the Navier-Stokes equations [53,74], wave equations [58,51,48,49,79,65,78], and second-order ODEs [61]. Moreover, second-derivative SBP operators can be used as building blocks in artificial dissipation operators and filtering procedures to stabilize numerical methods for hyperbolic problems [52,66,20,45,62]. ...
... To get an energy estimate, it suffices that the symmetric part of A, A + A T , is positive semi-definite, assuming that the BCs are enforced weakly using appropriate SATs. 1 In the context of FD-SBP operators, this generalization is necessary to construct compact second-derivative FD-SBP operators [50,53,47,12]. However, as we construct our second-derivative FSBP operators in a spectral element setting, we found no advantage in replacing D T 1 P D 1 with a more general A. Remark 2.4. ...
Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.
... Examples include finite difference (FD) [38,39,62,64], essentially non-oscillatory (ENO) and weighted ENO (WENO) [70,14,5], continuous Galerkin (CG) [1,2], discontinuous Galerkin (DG) [18,7], finite volume (FV) [51,52], flux reconstruction (FR) [34,61,57], and implicit time integration [54,40,60] methods. Second-derivative SBP operators have been the subject of significant research attention as well [46,43,12], for instance in the context of the Navier-Stokes equations [49,65], wave We obtained the numerical solutions using the FSBP-SAT scheme with polynomial ("poly") and exponential approximation spaces, P 2 = span{1, x, x 2 } and E 2 = span{1, x, e x }, respectively. See subsections 5.4 and 5.5 for more details. ...
... To get an energy estimate, it suffices that the symmetric part of A, A + A T , is positive semi-definite, assuming that the BCs are correctly enforced. 1 In the context of FD-SBP operators, this generalization is necessary to construct compact second-derivative FD-SBP operators [46,49,43,12]. However, as we construct our second-derivative FSBP operators in a spectral element setting, we found no advantage in replacing D T 1 P D 1 with a more general A. Remark 2.4. ...
Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. We showcase the superior performance of these non-polynomial FSBP operators over traditional polynomial-based operators for a suite of one- and two-dimensional problems, encompassing a boundary layer problem and the viscous Burgers' equation. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.
... The restriction operators simply restrict grid functions to the boundary grid point, e. g., e T l u = u(x 1 ). Note that although SBP discretizations may be generalized to grids in which the boundary points are excluded (so-called generalized SBP operators [7,9]), the operators presented herein include the boundary points. ...
... However, continuous approaches have also been explored [35,33]. This thesis focuses solely on first-derivative operators; however, note that one way to approximate a second derivative is by applying a first-derivative operator twice [20]. Examples of traditional and element-type SBP operators are given in the following subsections. ...
... Suppose that an artificial dissipation operator, Z, is constructed whose interior entries are defined by the coefficients resulting from the application of (2.23) at each interior mesh node. Reorganizing Z in terms of the variable coefficient b j+1/2 similar to the approach in [20] gives 24) where N denotes the number of grid nodes. Choosing N = 5 and selecting a boundary closure from [68] results in the following dissipation matrix for b j+1/2 = 1: ...
... 28) which is based on Definitions 5.3 and 5.4 of[20] and follows from the fact that the summation of two or morenegative semidefinite matrices results in a negative semidefinite matrix. The condition given by (2.28) holds for each M i matrix if and only if the eigenvalues of M i + M T i are all less than or equal to zero. ...
... damping of high frequency modes [12,13,27,30,31]. As with the wide-stencil operators, narrow-stencil second-derivative operators are coupled by simultaneous approximation terms (SATs) [6]. ...
... Although these SAT coefficients lead to adjoint consistent and stable discretizations, the analysis in [14] assumes positive definiteness of a component matrix of the discrete second derivative operator (i.e., M k in (2.4)). However, this condition is not satisfied by many narrow-stencil second-derivative operators in the literature, including those in [12,[27][28][29][30]. Furthermore, it is not straightforward how the theory extends to narrow-stencil generalized SBP operators which have one or more of the following characteristics: exclusion of one or both boundary nodes, non-repeating interior point operators, and non-uniform nodal distribution [12]. ...
... Although these SAT coefficients lead to adjoint consistent and stable discretizations, the analysis in [14] assumes positive definiteness of a component matrix of the discrete second derivative operator (i.e., M k in (2.4)). However, this condition is not satisfied by many narrow-stencil second-derivative operators in the literature, including those in [12,[27][28][29][30]. Furthermore, it is not straightforward how the theory extends to narrow-stencil generalized SBP operators which have one or more of the following characteristics: exclusion of one or both boundary nodes, non-repeating interior point operators, and non-uniform nodal distribution [12]. For a discussion on generalized SBP operators and a comparison of some of their properties with corresponding properties of classical SBP operators, we refer the reader to [10,12]. ...
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 2p 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.
... 1 Introduction Compared to wide-stencil 1 summation-by-parts (SBP) operators, explicitly formed narrow-stencil 2 second-derivative SBP operators provide smaller solution error, superior solution convergence rates, compact stencil width, and better damping of high frequency modes [29,30,26,12,13]. As with the wide-stencil operators, narrow-stencil secondderivative operators are coupled by simultaneous approximation terms (SATs) [6]. ...
... In a subsequent paper [14], Eriksson and Nordström used a variant of the approach in [13] to find a more general set of SAT coefficients. Although these SAT coefficients lead to adjoint consistent and stable discretizations in practice, the analysis in [14] assumes a condition that is not satisfied by many narrow-stencil second-derivative operators in the literature, including those in [29,28,26,12,27]. Furthermore, it is not straightforward how the theory extends to narrow-stencil generalized SBP operators which have one or more of the following characteristics: exclusion of one or both boundary nodes, non-repeating interior point operators, and non-uniform nodal distribution [12]. ...
... Although these SAT coefficients lead to adjoint consistent and stable discretizations in practice, the analysis in [14] assumes a condition that is not satisfied by many narrow-stencil second-derivative operators in the literature, including those in [29,28,26,12,27]. Furthermore, it is not straightforward how the theory extends to narrow-stencil generalized SBP operators which have one or more of the following characteristics: exclusion of one or both boundary nodes, non-repeating interior point operators, and non-uniform nodal distribution [12]. ...
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 $ 2p $ 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.
... Although the RV framework is presented in the context of SBP finite differences, the key elements for proving stability of (11) are the combination of SBP operators and stable boundary treatment. One might therefore consider using other types of SBP operators, for instance the generalized SBP operators presented in e.g., [37,5], or other methods for imposing boundary conditions, such as the projection method presented in Mattsson and Olsson [30]. ...
In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; (ii) first-order viscosity applied in regions with strong discontinuities; (iii) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.
... derivatives [14,15]. The operators satisfying the GSBP definition include a wide variety of operators with the following properties: ...
... The majority of problems currently solved using the SBP-SAT approach employ traditional SBP operators, in which a repeated interior point operator is utilized to approximate a given derivative on a uniform nodal distribution that includes both boundary nodes. However, recently, Del Rey Fernández et al. extended the SBP property to encompass a larger class of operators for first and second derivatives, including the following types of operators: operators that do not have a repeated interior point operator, operators that have nonuniform nodal distributions, and operators that do not include one or both boundary nodes [20,21]. These generalized summation-by-parts (GSBP) operators can potentially lead to flexible and efficient schemes that are arbitrarily high-order, discretely conservative, and provably stable, which motivates their further investigation and development. ...
... Suppose that an artificial dissipation operator, Z, is constructed whose interior entries are defined by the coefficients resulting from the application of Eq. (8) at each interior mesh node. Reorganizing Z in terms of the variable coefficient b j+1/2 similar to the approach in [21] gives ...
... which is based on Definitions 5.3 and 5.4 of [21] and follows from the fact that the summation of two or more negative semidefinite matrices results in a negative semidefinite matrix. The condition given by Eq. (13) holds for each M i matrix if and only if the eigenvalues of M i + M T i are all less than or equal to zero. ...
The summation-by-parts (SBP) property can be used to construct high-order provably stable numerical methods. A general framework is explored for deriving provably stable and conservative artificial dissipation operators for use with high-order traditional and element-type SBP operators on general nodal distributions, thus enabling the time stable and accurate solution of practical nonlinear problems, including those problems that contain variable coefficients, for example, aerodynamics problems involving the compressible Euler and Navier-Stokes equations. The basic premise is presented for scalar conservation laws and then extended to entropy stability for systems. Artificial dissipation operators for use with traditional SBP operators are constructed having 1st, 3rd, 5th, and 7th order accuracy on the interior achieved with minimum-width stencils that have ample flexibility in the derivation of novel accuracy-preserving boundary closures. Element-type dissipation operators are constructed on the Legendre-Gauss and Legendre-Gauss-Lobatto nodal distributions. The stability and accuracy properties of a suite of the constructed artificial dissipation operators are characterized in the numerical solution of the quasi-one-dimensional Euler equations applied to a converging-diverging nozzle.
... Interior penalties are known as simultaneous approximation terms (SATs) in the SBP literature [11]. Penalties for second-order PDEs have been well studied by both the SBP community [12][13][14][15] and the finite-element community (see the review [16] and the references therein). Nevertheless, multidimensional SBP-SAT discretizations introduce generalizations that have not, to the best of our knowledge, been considered in the either the DG or the SBP-SAT literature. ...
... [13], the nodes are usually assumed to lie on the interface. In [15], the authors consider tensor-product discretizations without nodes on the interfaces, but only the Baumann-Oden [18] penalty is investigated. Furthermore, adjoint consistency is rarely addressed in the SBP literature and has not been considered for operators whose nodes are strictly interior to the element. ...
... Left multiplying the strong-form discretization (15) by v T κ H κ and using (13), we arrive at ...
This work focuses on multidimensional summation-by-parts (SBP) discretizations of linear elliptic operators with variable coefficients. We consider a general SBP discretization with dense simultaneous approximation terms (SATs), which serve as interior penalties to enforce boundary conditions and inter-element coupling in a weak sense. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties. Based on these conditions, we generalize the modified scheme of Bassi and Rebay (BR2) and the symmetric interior penalty Galerkin (SIPG) method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.
