Stability and Functional Superconvergence of Narrow-Stencil Second-Derivative Generalized Summation-By-Parts Discretizations

TL;DR

This paper demonstrates that narrow-stencil second-derivative generalized SBP discretizations of diffusion problems are stable and achieve superconvergence of linear functionals at a rate of 2p, under certain smoothness and consistency conditions.

Contribution

The paper establishes stability conditions and superconvergence rates for narrow-stencil generalized SBP operators in diffusion discretizations, extending previous results to this specific operator class.

Findings

Superconvergence rate of 2p for linear functionals with narrow-stencil SBP operators.

Stability conditions for adjoint consistent narrow-stencil SBP discretizations.

Numerical verification using the 1D Poisson problem.

Abstract

We analyze the stability and functional superconvergence of discretizations of diffusion problems with the narrow-stencil second-derivative generalized summation-by-parts (SBP) operators coupled with simultaneous approximation terms (SATs). Provided that the primal and adjoint solutions are sufficiently smooth and the SBP-SAT discretization is primal and adjoint consistent, we show that linear functionals associated with the steady diffusion problem superconverge at a rate of $ 2p $ when a degree $ p+1 $ narrow-stencil or a degree $ p $ wide-stencil generalized SBP operator is used for the spatial discretization. Sufficient conditions for stability of adjoint consistent discretizations with the narrow-stencil generalized SBP operators are presented. The stability analysis assumes nullspace consistency of the second-derivative operator and the invertibility of the matrix approximating the first derivative at the element boundaries. The theoretical results are verified by numerical experiments with the one-dimensional Poisson problem.