Summation-by-parts approximations of the second derivative: Pseudoinverses of singular operators and revisiting the sixth order accurate narrow-stencil operator

TL;DR

This paper develops a pseudoinverse for singular finite difference operators approximating the second derivative, especially under Neumann boundary conditions, and improves sixth order accurate narrow-stencil operators through parameter analysis and numerical validation.

Contribution

It derives an explicit Moore-Penrose pseudoinverse for singular second derivative operators with Neumann boundaries and analyzes parameter choices to enhance sixth order accurate operators.

Findings

Derived a Moore-Penrose pseudoinverse for singular operators.

Identified conditions for the pseudoinverse's validity based on eigenvalues.

Numerical experiments demonstrate improved operator properties.

Abstract

We consider finite difference approximations of the second derivative, exemplified in Poisson's equation, the heat equation and the wave equation. The finite difference operators satisfy a summation-by-parts property, which mimics the integration-by-parts. Since the operators approximate the second derivative, they are singular by construction. To impose boundary conditions, these operators are modified using Simultaneous Approximation Terms. This makes the modified matrices non-singular, for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, when considering Neumann boundary conditions on both boundaries, the modified matrix is still singular. For such matrices, we have derived an explicit expression for the Moore-Penrose pseudoinverse, which can be used for solving elliptic problems and some time-dependent problems. The condition for this new pseudoinverse to be valid, is that the modified matrix does not have more than one zero eigenvalue. We have reconstructed the sixth order accurate narrow-stencil operator with a free parameter and show that more than one zero eigenvalue can occur. We have performed a detailed analysis on the free parameter to improve the properties of the second derivative operator. We complement the derivations by numerical experiments to demonstrate the improvements of the new second derivative operator.