DeepAI AI Chat
Log In Sign Up

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

by   Zelalem Arega Worku, et al.

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.


page 1

page 2

page 3

page 4


Inverses of SBP-SAT finite difference operators approximating the first and second derivative

The scalar, one-dimensional advection equation and heat equation are con...

On the Stability of IMEX Upwind gSBP Schemes for Linear Advection-Diffusion Equations

A fully discrete energy stability analysis is carried out for linear adv...

A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

This paper develops a unified general framework for designing convergent...

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

In this paper we propose and analyze an energy stable numerical scheme f...

Numerical simulation of inextensible elastic ribbons

Using dimensionally reduced models for the numerical simulation of thin ...