Efficient Numerical Evaluation of Triple Integral Using the Euler's Method and Richardson's Extrapolation
Shubhangini Gupta, Prashant Sharma, Tamal Pramanick
TL;DR
This paper tackles efficient numerical evaluation of a triple integral by recasting it as a third-order initial-value problem using the Leibniz and chain rules. It combines Euler's method as a simple baseline with Richardson extrapolation to systematically reduce discretization error, enabling higher-order accuracy without excessive computation. A model problem demonstrates that Richardson extrapolation converges faster than plain Euler and can achieve near-machine-precision accuracy under adaptively chosen step sizes. The approach offers a practical pathway for multidimensional integrals and can be extended to higher dimensions and applications in fluid dynamics, electromagnetics, and related fields.
Abstract
In this study, we employ Euler's method and Richardson's extrapolation to solve a triple integral, which is then transformed into a third-order initial value problem. Our objective is to resolve the computational challenges associated with triple integration by transforming it into an initial value problem. Euler's method is the fundamental numerical technique for approximating the solution, thereby establishing a baseline for accuracy. The precision of our computations is subsequently improved by employing Richardson's extrapolation to reduce errors systematically. This approach not only illustrates the adaptability of numerical methods in solving intricate mathematical problems, but it also emphasizes the significance of strategic error reduction techniques in enhancing computational outcomes. We present the efficacy of this method in solving triple integrals in an efficient manner through experimentation and analysis, thereby making a significant contribution to the fields of numerical computation and mathematical modeling.