A dual consistent finite difference method with narrow stencil second derivative operators

TL;DR

This paper develops a dual consistent finite difference method using narrow-stencil second derivative operators, enhancing boundary condition implementation for time-dependent PDEs with superconvergent properties.

Contribution

It generalizes dual consistency results to include a broader class of boundary conditions and narrow-stencil operators, supported by theoretical derivations and numerical experiments.

Findings

Superconvergent functional output achieved with dual consistent discretizations

Generalization to narrow-stencil second derivative operators

Numerical experiments confirm theoretical results

Abstract

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP-SAT). Recently it was shown that SBP-SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.