Towards symmetric discretization schemes via weak boundary conditions

TL;DR

This paper proposes a novel approach to symmetric discretization schemes for classical one-dimensional problems, avoiding doubling issues by weakly imposing boundary conditions and using regularized summation-by-parts operators.

Contribution

It introduces a new method for symmetric discretization that leverages weak boundary conditions and regularized operators, addressing doubling problems in gauge theory discretizations.

Findings

Successfully formulated symmetric schemes avoiding doubling

Applied methods to classical initial value problems with second order derivatives

Demonstrated improved discretization accuracy and stability

Abstract

The Szymanzik improvement program for gauge theories is most commonly implemented using forward finite difference corrections to the Wilson action. Central symmetric schemes naively applied, suffer from a doubling of degrees of freedom, identical to the well known fermion doubling phenomenon. And while adding a complex Wilson term remedies the problem for fermions, it does not easily transfer to real-valued gauge fields. In this talk I report on recent progress in formulating symmetric discretization schemes for classical actions of simple one-dimensional problems. They avoid doubling by exploiting the weak imposition of initial/boundary conditions. Inspired by recent work in the field of numerical analysis of partial differential equations, I construct a regularized summation-by-parts finite difference operator using boundary data based on affine coordinates. Application to a classical initial value problems with second order derivatives are presented.