Jason E. Hicken

Jason E. Hicken, David W. Zingg

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.

Jianfeng Yan, Jared Crean, Jason E. Hicken

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.

David C. Del Rey Fernández, Jason E. Hicken, David W. Zingg

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.

Jason E. Hicken, David C. Del Rey Fernández, David W. Zingg

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.