[Paper Review] On the exact discretization of the classical harmonic oscillator equation

Jan L. Cieśliński|arXiv (Cornell University)|Nov 18, 2009

Numerical methods for differential equationsMathematics37 references30 citations

TL;DR

This paper presents exact discretizations of the classical harmonic oscillator equation—both homogeneous and inhomogeneous, one-dimensional and multidimensional—by deriving discrete analogues that preserve the energy integral exactly. It introduces a family of geometric integrators based on symplectic, energy-preserving, and time-reversible difference schemes, with key results showing that the discrete energy is conserved for three equivalent formulations, and demonstrates applications in wave equations and locally exact numerical schemes.

ABSTRACT

We discuss the exact discretization of the classical harmonic oscillator equation (including the inhomogeneous case and multidimensional generalizations) with a special stress on the energy integral. We present and suggest some numerical applications.

Motivation & Objective

• To develop exact finite difference schemes for the classical harmonic oscillator equation that preserve the energy integral exactly.
• To generalize exact discretization to inhomogeneous, damped, and multidimensional cases, including the Kepler problem via Kustaanheimo-Stiefel transformation.
• To establish connections between exact discretization and geometric numerical integration, particularly in preserving symplecticity and time-reversibility.
• To provide a foundation for constructing locally exact numerical schemes by modifying discrete gradient methods using exact discretization principles.
• To explore applications in partial differential equations, such as the linearized wave equation, where the numerical frequency matches the physical frequency.

Proposed method

• Derives exact discretization of the harmonic oscillator equation using a second-order linear difference equation: $ x_{n+1} - 2\cos(\omega\varepsilon)x_n + x_{n-1} = 0 $, which exactly reproduces the continuous solution at discrete time steps.
• Introduces a discrete velocity approximation $ v_n = \frac{x_{n+1} - x_n \cos(\omega\varepsilon)}{\varepsilon \sin(\omega\varepsilon)/\omega} $, ensuring consistency with the continuous velocity.
• Proposes three equivalent forms of the discrete energy integral: $ E_n^{(0)} $, $ E_n^{(1)} $, and $ E_n^{(2)} $, all of which are conserved under the exact discretization.
• Constructs a family of geometric integrators via the map $ x_{n+1} - \gamma x_n + x_{n-1} = 0 $, $ p_n = \alpha x_{n+1} - \beta x_n $, with conditions on $ \alpha, \beta, \gamma $ ensuring symplecticity, energy preservation, and time-reversibility.
• Applies exact discretization to the Kepler problem by transforming it into a 4D harmonic oscillator, enabling exact trajectory preservation.
• Introduces a locally exact modification of the discrete gradient scheme by replacing the fixed time step $ \varepsilon $ with a variable $ \delta_n $, chosen such that the scheme becomes exact for the linearized system around a point.

Experimental results

Research questions

• RQ1 How can the classical harmonic oscillator equation be exactly discretized such that the discrete solution matches the continuous solution at each time step?
• RQ2 What are the equivalent formulations of the exact discretization that preserve key physical invariants like energy and symplectic structure?
• RQ3 Can exact discretization be extended to inhomogeneous, damped, and multidimensional harmonic oscillators while preserving geometric properties?
• RQ4 How can exact discretization be used to construct locally exact numerical schemes that are exact for small oscillations around a given point?
• RQ5 What are the implications of exact discretization for solving partial differential equations, such as the linearized wave equation?

Key findings

• The exact discretization $ x_{n+1} - 2\cos(\omega\varepsilon)x_n + x_{n-1} = 0 $ exactly reproduces the continuous harmonic oscillator solution at discrete times $ t_n = \varepsilon n $, with $ x_n = x(\varepsilon n) $.
• Three discrete energy forms—$ E_n^{(0)} $, $ E_n^{(1)} $, and $ E_n^{(2)} $—are exactly conserved under the exact discretization, with $ E_n^{(2)} $ being the most natural discrete analogue of the continuous energy $ \frac{1}{2}v_n^2 + \frac{1}{2}\omega^2 x_n^2 $.
• The discrete velocity $ v_n = \frac{x_{n+1} - x_n \cos(\omega\varepsilon)}{\varepsilon \sin(\omega\varepsilon)/\omega} $ ensures that the energy $ \frac{1}{2}v_n^2 + \frac{1}{2}\omega^2 x_n^2 $ is conserved exactly.
• The family of maps defined by $ x_{n+1} - \gamma x_n + x_{n-1} = 0 $, $ p_n = \alpha x_{n+1} - \beta x_n $, is symplectic, energy-preserving, and time-reversible when the coefficients satisfy $ \alpha(-\varepsilon) = -\alpha(\varepsilon) $, $ \gamma(\varepsilon) = \frac{\beta(\varepsilon) - \beta(-\varepsilon)}{\alpha(\varepsilon)} $.
• For the harmonic oscillator with constant force, the locally exact scheme is obtained by setting $ \delta_n = \frac{2}{\omega} \tan(\omega\varepsilon/2) $, ensuring exactness for the linearized system around a point.
• The exact discretization of the wave equation via Fourier transformation leads to a scheme where the numerical frequency matches the physical frequency $ \omega = \sqrt{k^2 + a^2} $, resulting in exact group velocity and energy transport.