Code Verification for Practically Singular Equations

TL;DR

This paper explores advanced methods for verifying code accuracy in electric-field integral equations, especially addressing challenges posed by singular integrals and numerical errors, and proposes a more sensitive error detection approach.

Contribution

It introduces a reformulated optimization-based approach to better detect subtle coding errors in practically singular systems of equations.

Findings

Discretization error from the optimal solution is more sensitive than truncation error.

Inserting the exact solution cannot detect certain coding errors.

The proposed method improves code verification sensitivity.

Abstract

The method-of-moments implementation of the electric-field integral equation (EFIE) yields many code-verification challenges due to the various sources of numerical error and their possible interactions. Matters are further complicated by singular integrals, which arise from the presence of a Green's function. To address these singular integrals, an approach was previously presented wherein both the solution and Green's function are manufactured. Because the arising equations are poorly conditioned, they are reformulated as a set of constraints for an optimization problem that selects the solution closest to the manufactured solution. In this paper, we demonstrate how, for such practically singular systems of equations, computing the truncation error by inserting the exact solution into the discretized equations cannot detect certain orders of coding errors. On the other hand, the discretization error from the optimal solution is a more sensitive metric that can detect orders less than those of the expected convergence rate.