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.
A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
ETSIAE-UPM - School of Aeronautics, Universidad Politécnica de Madrid（马德里理工大学航空学院）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Computational Science Research Center, San Diego State University（圣地亚哥州立大学计算科学研究中心）
Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes equations with variable density and artificial compressibility
A Non-stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces
A finite-difference summation-by-parts, conditionally stable partitioned algorithm for conjugate heat transfer problems
University of Waterloo（滑铁卢大学）；Ansys Canada Inc.（安世加拿大公司）；the University of British Columbia（不列颠哥伦比亚大学）；Institute of Mechanical and Electrical Engineering, University of Southern Denmark（南丹麦大学机电工程研究所）
Summation-by-parts approximations of the second derivative: Pseudoinverses of singular operators and revisiting the sixth order accurate narrow-stencil operator
A Flux-Differencing Formula for Split-Form Summation By Parts Discretizations of Non-Conservative Systems: Applications to Subcell Limiting for magneto-hydrodynamics
University of Cologne（科隆大学）；Universidad Politécnica de Madrid（马德里理工大学）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Technical University of Munich（慕尼黑工业大学）；University of Waterloo（滑铁卢大学）
A high order accurate and energy stable continuous Galerkin framework on summation-by-parts form for the incompressible Navier-Stokes equations
InCFD Research Group, Department of Mechanical Engineering, University of Cape Town（开普敦大学机械工程系）；Department of Mathematical Sciences, Applied Mathematics Division, Stellenbosch University（斯泰伦博斯大学应用数学系）；Department of Mathematics, Applied Mathematics, Linköping University（林雪平大学应用数学系）；Department of Mathematics and Applied Mathematics, University of Johannesburg（约翰内斯堡大学应用数学系）
