ContEvol formalism: numerical methods based on Hermite spline optimization

TL;DR

ContEvol formalism introduces implicit numerical methods based on Hermite spline optimization that are nearly symplectic, offering improved accuracy and versatility for solving differential equations in physics and quantum mechanics.

Contribution

This paper presents the ContEvol formalism, a novel family of implicit, Hermite spline-based numerical methods that outperform traditional Runge--Kutta methods in accuracy and applicability.

Findings

ContEvol methods achieve lower errors than Runge--Kutta in harmonic oscillator simulations.

First-order ContEvol maintains high accuracy in celestial mechanics problems with $ ext{O}(h^5)$ deviation.

ContEvol effectively solves boundary value and eigenvalue problems in quantum mechanics.

Abstract

We present the ContEvol (continuous evolution) formalism, a family of implicit numerical methods which only need to solve linear equations and are almost symplectic. Combining values and derivatives of functions, ContEvol outputs allow users to recover full history and render full distributions. Using the classic harmonic oscillator as a prototype case, we show that ContEvol methods lead to lower-order errors than two commonly used Runge--Kutta methods. Applying first-order ContEvol to simple celestial mechanics problems, we demonstrate that deviation from equation(s) of motion of ContEvol tracks is still $\mathcal{O}(h^5)$ ($h$ is the step length) by our definition. Numerical experiments with an eccentric elliptical orbit indicate that first-order ContEvol is a viable alternative to classic Runge--Kutta or the symplectic leapfrog integrator. Solving the stationary Schr\"odinger equation in quantum mechanics, we manifest ability of ContEvol to handle boundary value or eigenvalue problems. Important directions for future work, including mathematical foundations, higher dimensions, and technical improvements, are discussed at the end of this article.