arXiv subjects

Jason E. Hicken

Publications and source records attributed to Jason E. Hicken.

10 recordsLinked to original sources

arXiv2024

Very high-order symmetric positive-interior quadrature rules on triangles and tetrahedra

We present novel fully-symmetric quadrature rules with positive weights and strictly interior nodes of degrees up to 84 on triangles and 40 on tetrahedra. Initial guesses for solving the nonlinear systems of equations needed to derive quadrature rules are generated by forming tensor-product structures on quadrilateral/hexahedral subdomains of the simplices using the Legendre-Gauss nodes on the first half of the line reference element. In combination with a methodology for node elimination, these initial guesses lead to the development of highly efficient quadrature rules, even for very high polynomial degrees. Using existing estimates of the minimum number of quadrature points for a given degree, we show that the derived quadrature rules on triangles and tetrahedra are more than 95% and 80% efficient, respectively, for almost all degrees. The accuracy of the quadrature rules is demonstrated through numerical examples.

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arXiv2024

Tensor-Product Split-Simplex Summation-By-Parts Operators

We present an approach to construct efficient sparse summation-by-parts (SBP) operators on triangles and tetrahedra with a tensor-product structure. The operators are constructed by splitting the simplices into quadrilateral or hexahedral subdomains, mapping tensor-product SBP operators onto the subdomains, and assembling back using a continuous-Galerkin-type procedure. These tensor-product split-simplex operators do not have repeated degrees of freedom at the interior interfaces between the split subdomains. Furthermore, they satisfy the SBP property by construction, leading to stable discretizations. The accuracy and sparsity of the operators substantially enhance the efficiency of SBP discretizations on simplicial meshes. The sparsity is particularly important for entropy-stable discretizations based on two-point flux functions, as it reduces the number of two-point flux computations. We demonstrate through numerical experiments that the operators exhibit efficiency surpassing that of the existing dense multidimensional SBP operators by more than an order of magnitude in many cases. This superiority is evident in both accuracy per degree of freedom and computational time required to achieve a specified error threshold.

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arXiv2023

Quadrature Rules on Triangles and Tetrahedra for Multidimensional Summation-By-Parts Operators

Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-$ \mathsf{E} $ operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However, there is a limited number of such operators, and those currently in existence often have a relatively high node count for a given polynomial order due to a scarcity of suitable quadrature rules. We present several new symmetric positive-weight quadrature rules on triangles and tetrahedra that are suitable for construction of diagonal-$ \mathsf{E} $ SBP operators. For triangles, quadrature rules of degree one through twenty with facet nodes that correspond to the Legendre-Gauss-Lobatto (LGL) and Legendre-Gauss (LG) quadrature rules are derived. For tetrahedra, quadrature rules of degree one through ten are presented along with the corresponding facet quadrature rules. All of the quadrature rules are provided in a supplementary data repository. The quadrature rules are used to construct novel SBP diagonal-$ \mathsf{E} $ operators, whose accuracy and maximum timestep restrictions are studied numerically.

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arXiv2021

Entropy-stable discontinuous Galerkin difference methods for hyperbolic conservation laws

The paper describes the construction of entropy-stable discontinuous Galerkin difference (DGD) discretizations for hyperbolic conservation laws on unstructured grids. The construction takes advantage of existing theory for entropy-stable summation-by-parts (SBP) discretizations. In particular, the paper shows how DGD discretizations -- both linear and nonlinear -- can be constructed by defining the SBP trial and test functions in terms of interpolated DGD degrees of freedom. In the case of entropy-stable discretizations, the entropy variables rather than the conservative variables must be interpolated to the SBP nodes. A fully-discrete entropy-stable scheme is obtained by adopting the relaxation Runge-Kutta version of the midpoint method. In addition, DGD matrix operators for the first derivative are shown to be dense-norm SBP operators. Numerical results are presented to verify the accuracy and entropy-stability of the DGD discretization in the context of the Euler equations. The results suggest that DGD and SBP solution errors are similar for the same number of degrees of freedom. Finally, an investigation of the DGD spectra shows that spectral radius is relatively insensitive to discretization order; however, the high-order methods do suffer from the linear instability reported for other entropy-stable discretizations.

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arXiv2020

Entropy-stable, high-order summation-by-parts discretizations without interface penalties

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. Furthermore, LPS is able to take advantage of the additional nodes required by degree $2p$ diagonal-norms, resulting in an element-local stabilization with a bounded spectral radius. An entropy-stable version of LPS is easily obtained by applying the projection on the entropy variables. Numerical experiments with the linear-advection and Euler equations demonstrate the accuracy, efficiency, and robustness of the stabilized discretizations, and the continuous approach compares favorably with the more common discontinuous SBP methods.

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arXiv2017

Error-tolerant Multisecant Method for Nonlinearly Constrained Optimization

We present a derivative-based algorithm for nonlinearly constrained optimization problems that is tolerant of inaccuracies in the data. The algorithm solves a semi-smooth set of nonlinear equations that are equivalent to the first-order optimality conditions, and it is matrix-free in the sense that it does not require the explicit Lagrangian Hessian or Jacobian of the constraints. The solution method is quasi-Newton, but rather than approximating only the Hessian or constraint Jacobian, the Jacobian of the entire nonlinear set of equations is approximated using a multisecant method. We show how preconditioning can be incorporated into the multisecant update in order to improve the performance of the method. For nonconvex problems, we propose a simple modification of the secant conditions to regularize the Hessian. Numerical experiments suggest that the algorithm is a promising alternative to conventional gradient-based algorithms, particularly when errors are present in the data.

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arXiv2016

Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

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.

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arXiv2016

Simultaneous Approximation Terms for Multi-Dimensional Summation-by-Parts Operators

This paper is concerned with the accurate, conservative, and stable imposition of boundary conditions and inter-element coupling for multi-dimensional summation-by-parts (SBP) finite-difference operators. More precisely, the focus is on diagonal-norm SBP operators that are not based on tensor products and are applicable to unstructured grids composed of arbitrary elements. We show how penalty terms --- simultaneous approximation terms (SATs) --- can be adapted to discretizations based on multi-dimensional SBP operators to enforce boundary and interface conditions. A general SAT framework is presented that leads to conservative and stable discretizations of the variable-coefficient advection equation. This framework includes the case where there are no nodes on the boundary of the SBP element at which to apply penalties directly. This is an important generalization, because elements analogous to Legendre-Gauss collocation, \ie without boundary nodes, typically have higher accuracy for the same number of degrees of freedom. Symmetric and upwind examples of the general SAT framework are created using a decomposition of the symmetric part of an SBP operator; these particular SATs enable the pointwise imposition of boundary and inter-element conditions. We illustrate the proposed SATs using triangular-element SBP operators with and without nodes that lie on the boundary. The accuracy, conservation, and stability properties of the resulting SBP-SAT discretizations are verified using linear advection problems with spatially varying divergence-free velocity fields.

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arXiv2015

Multidimensional Summation-By-Parts Operators: General Theory and Application to Simplex Elements

Summation-by-parts (SBP) finite-difference discretizations share many attractive properties with Galerkin finite-element methods (FEMs), including time stability and superconvergent functionals; however, unlike FEMs, SBP operators are not completely determined by a basis, so the potential exists to tailor SBP operators to meet different objectives. To date, application of high-order SBP discretizations to multiple dimensions has been limited to tensor product domains. This paper presents a definition for multi-dimensional SBP finite-difference operators that is a natural extension of one-dimensional SBP operators. Theoretical implications of the definition are investigated for the special case of a diagonal norm (mass) matrix. In particular, a diagonal-norm SBP operator exists on a given domain if and only if there is a cubature rule with positive weights on that domain and the polynomial-basis matrix has full rank when evaluated at the cubature nodes. Appropriate simultaneous-approximation terms are developed to impose boundary conditions weakly, and the resulting discretizations are shown to be time stable. Concrete examples of multi-dimensional SBP operators are constructed for the triangle and tetrahedron; similarities and differences with spectral-element and spectral-difference methods are discussed. An assembly process is described that builds diagonal-norm SBP operators on a global domain from element-level operators. Numerical results of linear advection on a doubly periodic domain demonstrate the accuracy and time stability of the simplex operators.

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arXiv2011

Summation-By-Parts Operators and High-Order Quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. The accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. The accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; for example, the discrete norm accurately approximates the $L^{2}$ norm for functions, and multi-dimensional SBP discretizations accurately mimic the divergence theorem.

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