On the Partial Analytical Solution of the Kirchhoff Equation

TL;DR

This paper presents a hybrid analytical and numerical approach to solve the (1+1)-dimensional Kirchhoff system, improving accuracy and stability over purely numerical methods by using symmetry analysis and differential decomposition.

Contribution

The authors develop a combined analytical-numerical scheme using Lie symmetry and Thomas decomposition to efficiently solve the Kirchhoff equation with enhanced stability.

Findings

Reduced numerical instability in simulations

Improved solution accuracy for the Kirchhoff system

Effective simulation of a cilia carpet model

Abstract

We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.