Schwartz duality for singularly perturbed nonlinear differential equations with Chebyshev spectral method

TL;DR

This paper introduces a novel Schwartz duality-based spectral method that effectively solves nonlinear singularly perturbed differential equations with Dirac delta functions, eliminating Gibbs oscillations and improving accuracy.

Contribution

It extends Schwartz duality techniques to nonlinear problems using a modified projection method with a discrete Heaviside derivative.

Findings

Eliminates Gibbs oscillations in nonlinear problems

Achieves uniform error reduction

Demonstrates superior accuracy over traditional methods

Abstract

Singularly perturbed differential equations with a Dirac delta function yield discontinuous solutions. Therefore, careful consideration is required when using numerical methods to solve these equations because of the Gibbs phenomenon. A remedy based on the Schwartz duality has been proposed, yielding superior results without oscillations. However, this approach has been limited to linear problems and still suffers from the Gibbs phenomenon for nonlinear problems. In this note, we propose a consistent yet simple approach based on Schwartz duality that can handle nonlinear problems. Our proposed approach utilizes a modified direct projection method with a discrete derivative of the Heaviside function, which directly approximates the Dirac delta function. This proposed method effectively eliminates Gibbs oscillations without the need for traditional regularization and demonstrates uniform error reduction.