[Paper Review] A very accurate method to approximate discontinuous functions with a finite number of discontinuities

Eduardo Stella, Celso L. Ladera+1 more|arXiv (Cornell University)|Jan 19, 2016

Iterative Methods for Nonlinear Equations12 references3 citations

TL;DR

This paper presents a highly accurate, analytic method to approximate functions with a finite number of discontinuities by using rational-argument hyperbolic tangent functions as smooth connecting functions. The approach avoids the Gibbs phenomenon, achieving relative errors as low as 10⁻¹⁴ even near discontinuities, and solves a linear system to determine coefficients from partition functions, enabling precise convergence at jump points.

ABSTRACT

A simple and very accurate method to approximate a function with a finite number of discontinuities is presented. This method relies on hyperbolic tangent functions of rational arguments as connecting functions at the discontinuities, each argument being the reciprocal of Newton binomials that depend on the abscissae that define the domain of the discontinuous function and upon the abscissae of discontinuities. Our approximants take the form of linear combinations of such hyperbolic tangent functions with coefficients that are obtained by solving a linear system of inhomogeneous equations whose righthand sides are the partition functions that define the given discontinuous function. These approximants are analytic, and being free from the Gibbs phenomenon certainly converge at the discontinuity points much better than other known approximants to discontinuous functions, typical relative errors being of the order of 10-14 even when as close as 10-12 to the discontinuity points. Moreover, they can be readily scaled to larger intervals. Our method is here illustrated with a representative set of discontinuous mathematical physics functions, and by studying the dynamics of an oscillator subjected to a discontinuous force, but it can be applied to important cases of discontinuous functions in physics, mathematics, engineering and physical chemistry.

Motivation & Objective

• To develop a method that accurately approximates discontinuous functions with a finite number of jump discontinuities.
• To eliminate the Gibbs phenomenon, which plagues traditional Fourier-based approximations near discontinuities.
• To provide analytic approximants that converge reliably at discontinuity points.
• To enable scalable approximation over larger intervals without loss of accuracy.
• To offer a practical and numerically robust approach applicable to physics, engineering, and mathematical modeling.

Proposed method

• The method constructs approximants as linear combinations of hyperbolic tangent functions with rational arguments derived from Newton binomials.
• Each hyperbolic tangent function is centered at a discontinuity point and modulated by the reciprocal of a Newton binomial depending on domain boundaries and discontinuity locations.
• Coefficients in the linear combination are determined by solving a system of inhomogeneous linear equations, with right-hand sides given by the partition functions defining the discontinuous function.
• The resulting approximants are analytic and smooth across the entire domain, including at discontinuity points.
• The method ensures convergence at discontinuities with high precision, even when approaching within 10⁻¹² of the jump.
• The formulation allows straightforward scaling to larger intervals while preserving accuracy.

Experimental results

Research questions

• RQ1 Can a smooth, analytic approximation be constructed for a discontinuous function with finite jumps that converges accurately at the discontinuity points?
• RQ2 How can the Gibbs phenomenon be avoided in approximating functions with jump discontinuities?
• RQ3 What functional form and parameterization yield high-accuracy approximations even near discontinuities?
• RQ4 Can the method be generalized to functions with multiple discontinuities and scaled to larger domains?
• RQ5 What level of accuracy can be achieved in relative error when approaching discontinuity points?

Key findings

• The method achieves relative errors as low as 10⁻¹⁴ even when approaching discontinuity points within 10⁻¹², demonstrating exceptional numerical precision.
• The approximants are analytic and converge at discontinuity points, unlike traditional Fourier series that exhibit the Gibbs phenomenon.
• The use of rational-argument hyperbolic tangent functions enables smooth transitions across jumps while preserving high accuracy.
• The linear system of equations for coefficient determination is well-conditioned and numerically stable, ensuring reliable computation.
• The method is scalable and applicable to functions in mathematical physics, engineering, and physical chemistry with discontinuous behavior.
• Illustrative examples, including an oscillator under a discontinuous force, confirm the method’s robustness and high accuracy in dynamic systems.