A Fast Tree Algorithm for Electric Field Calculation in Electrical Discharge Simulations

TL;DR

This paper introduces a fast tree algorithm that significantly accelerates electric field calculations in electrical discharge simulations, reducing computational complexity from quadratic to near-linear, with controllable accuracy.

Contribution

The paper presents a novel tree algorithm that improves computational efficiency for electric field calculations in discharge simulations, with rigorous error control and validation.

Findings

Reduces computation time from O(N^2) to O(N log N)

Error decays exponentially with truncation terms

Validated accuracy and efficiency through numerical examples

Abstract

The simulation of electrical discharges has been attracting a great deal of attention. In such simulations, the electric field computation dominates the computational time. In this paper, we propose a fast tree algorithm that helps to reduce the time complexity from $O(N^2)$ (from using direct summation) to $O(N\log N)$. The implementation details are discussed and the time complexity is analyzed. A rigorous error estimation shows the error of the tree algorithm decays exponentially with the number of truncation terms and can be controlled adaptively. Numerical examples are presented to validate the accuracy and efficiency of the algorithm.