Generalized Summation-by-parts Operators for First and Second Derivatives

Generalized Summation-by-Parts Operators for First and Second Derivatives David César Del Rey Fernández Doctor of Philosophy Graduate Department of Institute for Aerospace Studies University of Toronto 2015 Higher-order methods represent an attractive means of efficiently solving partial differential equations (PDEs) for certain classes of problems. The theory of classical finite-difference (FD) summation-by-parts (SBP) operators with weak imposition of boundary conditions using simultaneous approximation terms (SATs) is extended in several new directions. The SBP-SAT approach is advantageous as it leads to provably consistent, conservative, and stable higher-order discretizations, gives a straightforward means to develop numerical boundary conditions and inter-block coupling, and results in efficient parallel schemes with constant communication overhead, regardless of the order of the scheme. We develop a framework for generalized SBP (GSBP) operators that extends the theory of classical FD-SBP operators to operators with one or more of the following characteristics: i) non-repeating interior operator, ii) nonuniform nodal distribution in the computational domain, and iii) operators that exclude one or both boundary nodes. Necessary and sufficient conditions for the existence of first-derivative GSBP operators are derived. For diagonalnorm operators, we show that if a quadrature rule with positive weights exists, then an associated GSBP operator is guaranteed to exist. This reduces the search for diagonal-norm GSBP operators to the search for quadrature rules with positive weights. Furthermore, we prove that dense-norm GSBP operators on n distinct nodes of up to order n−1 always exist. The framework enables one to show that many known operators have the SBP property and gives a straightforward methodology for the construction of novel operators. We extend the GSBP framework to include approximations to the second derivative with a variable coefficient and develop GSBP operators more accurate than the application of the first-derivative operator twice that lead to stable discretizations of PDEs with crossii derivative terms. We propose a novel decomposition of classical FD-SBP operators that leads to efficient implementations and simplifies the construction of Jacobian matrices. We show how to construct SATs, derive various novel operators, and demonstrate that they have preferential error characteristics relative to classical FD-SBP operators in the context of the linear convection and linear convection-diffusion equations.