Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations

Abstract

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.

Summation-by-Parts Operators with Minimal Dispersion Error for Accurate and Efficient Flow Calculations

“…SBP operators [9][10][11][12]23 augmented with Simultaneous Approximation Terms (SAT) 24 that weakly enforce boundary conditions (the SBP-SAT technique) allow for stable and high order discretisations of systems of partial differential equations if well posed boundary conditions are available.…”

Aerodynamic Shape Optimization of a Box-Wing Regional Aircraft Based on the Reynolds-Averaged Navier-Stokes Equations

“…These are used in conjunction with simultaneous-approximation terms (SATs) which enforce boundary conditions and inter-block coupling weakly. Together, SBP-SAT operators provide excellent numerical stability properties and efficient parallel performance, while only requiring C 0 mesh continuity at block interfaces [75]; recall from Section 2.1.1 that this condition is met by the definition of the knot vector. To aid in stabilizing the solution around shocks, a pressure sensor is used to control the activation of artificial dissipation.…”

Summation-by-Parts operators with minimal dispersion error for coarse grid flow calculations

“…Augmented with Simultaneous Approximation Terms (SAT) [17] that weakly enforce boundary conditions, the SBP-SAT technique allows for stable and high order discretisations of partial differential equations if well posed boundary conditions are available. For reviews of the SBP-SAT technique and its development, see [13,18] and the references therein.…”