Evaluating a double integral using Euler's method and Richardson extrapolation

TL;DR

The paper addresses evaluating a double integral by recasting it as a second-order initial-value problem and solving with Euler's method augmented by Richardson extrapolation to achieve high accuracy. It provides a concrete numerical example where the method attains near machine-precision accuracy and demonstrates how the framework yields an error curve for a cubature rule such as Simpson's rule. A key contribution is the explicit adaptive-step strategy that uses the Richardson error coefficient to meet a user-defined tolerance, along with the ability to quantify and reduce cubature errors via the $C(x)$ correction term. The approach offers practical benefits in efficiency and error control, with potential extensions to higher dimensions, subinterval partitioning, and general outer limits.

Abstract

We transform a double integral into a second-order initial value problem, which we solve using Euler's method and Richardson extrapolation. For an example we consider, we achieve accuracy close to machine precision (1e-15). We also use the algorithm to determine the error curve for a Simpson cubature rule.