[Paper Review] Trigonometric Splines for Oscillator Simulation

Kai Bittner, Hans-Georg Brachtendorf|arXiv (Cornell University)|Apr 26, 2016

Advanced Measurement and Metrology Techniques5 citations

TL;DR

This paper proposes trigonometric splines to eliminate numerical damping in oscillator simulations, which otherwise causes artificial energy loss and unreliable results. By replacing polynomial B-splines with trigonometric B-splines, the method preserves low-frequency oscillations exactly while maintaining stability and adaptability, enabling accurate long-term simulation of autonomous oscillators like the 3MHz Colpitz-Quartz oscillator.

ABSTRACT

We investigate the effects of numerical damping for oscillator simulation with spline methods. Numerical damping results in an artificial loss of energy and leads therefore to unreliable results in the simulation of autonomous systems, as e.g.\ oscillators. We show that the negative effects of numerical damping can be eliminated by the use of trigonometric splines. This will be in particular important for spline based adaptive methods.

Motivation & Objective

• To address numerical damping in spline-based simulations of oscillators, which artificially reduces energy and compromises accuracy in autonomous systems.
• To investigate why classical polynomial splines induce numerical damping, especially at the fundamental frequency, due to derivative approximation errors.
• To develop a spline method that preserves low-frequency oscillations exactly while still damping high-frequency numerical noise.
• To demonstrate that trigonometric splines provide exact derivative representation for low frequencies, eliminating artificial energy loss.
• To enable reliable adaptive grid methods by ensuring consistent numerical behavior across varying grid resolutions.

Proposed method

• Replace polynomial B-splines with trigonometric B-splines defined via a recursive sine-based formula that maintains compact support and smoothness.
• Use a collocation method on uniform grids where the function and its derivative are approximated at shifted grid points using trigonometric spline basis functions.
• Apply the discrete Fourier transform to analyze the amplification factor of the derivative approximation, defined as ψ̃_m,h(x,ξ) = (∂/∂x φ̃_m,h(x,ξ)) / φ̃_m,h(x,ξ).
• Leverage the property that trigonometric splines of odd order m=2μ+1 exactly represent complex exponentials e^{2πikt} for |k|<μ, ensuring exact derivative computation for low frequencies.
• Ensure stability and accuracy by choosing collocation parameters σ ∈ (−1/2, 0) to avoid singularities and maintain favorable damping behavior.
• Validate the method by simulating a 3MHz Colpitz-Quartz oscillator and comparing results with classical polynomial splines.

Experimental results

Research questions

• RQ1 Can trigonometric splines eliminate numerical damping in oscillator simulations while preserving stability?
• RQ2 Why do classical polynomial splines cause artificial energy loss in oscillators, and how does this depend on grid resolution and collocation point placement?
• RQ3 Can trigonometric splines exactly represent the fundamental frequency and its derivative, thus avoiding damping of the physical signal?
• RQ4 How does the choice of collocation parameter σ affect numerical damping and stability in trigonometric spline collocation?
• RQ5 Can trigonometric splines be extended to adaptive, non-uniform grids without introducing inconsistent damping effects?

Key findings

• Trigonometric splines of odd order m=3,5,… exactly represent low-frequency sinusoids, ensuring no numerical damping for the fundamental frequency.
• The derivative amplification factor ψ̃_m,h(σ,ξ) for trigonometric splines equals 2πik for |k|<μ, confirming exact derivative computation for low frequencies.
• For σ = −1/4, the classical polynomial spline method shows significant damping of the fundamental frequency, while trigonometric splines eliminate this effect.
• The simulation results using trigonometric splines match the expected amplitude within approximation error, confirming energy preservation.
• Trigonometric splines maintain stability and avoid ringing artifacts that occur with positive σ in polynomial splines, which lack high-frequency damping.
• The method is extendable to non-uniform grids used in adaptive methods, preventing inconsistent damping across different grid regions.