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

被引:254
作者
Fernandez, David C. Del Rey [1 ]
Hicken, Jason E. [2 ]
Zingg, David W. [1 ]
机构
[1] Univ Toronto, Inst Aerosp Studies, Toronto, ON M3H 5T6, Canada
[2] Rensselaer Polytech Inst, Dept Mech Aerosp & Nucl Engn, Troy, NY 12180 USA
关键词
Summation-by-parts operators; Finite-difference methods; Simultaneous approximation terms; Boundary closures; Numerical boundary schemes; Higher-order methods; Stability; Energy-method; ORDER FINITE-DIFFERENCE; NAVIER-STOKES EQUATIONS; DISCONTINUOUS GALERKIN METHODS; NONLINEAR CONSERVATION-LAWS; CONJUGATE HEAT-TRANSFER; BOUNDARY-CONDITIONS; WAVE-PROPAGATION; GENERALIZED FRAMEWORK; STRICT STABILITY; ERROR ESTIMATION;
D O I
10.1016/j.compfluid.2014.02.016
中图分类号
TP39 [计算机的应用];
学科分类号
081203 ; 0835 ;
摘要
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. (c) 2014 Elsevier Ltd. All rights reserved.
引用
收藏
页码:171 / 196
页数:26
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