Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

TL;DR

This paper introduces error analysis techniques for numerical solutions of nonlinear partial differential equations, emphasizing stability and accuracy, with illustrative examples and open research questions suitable for undergraduates.

Contribution

It presents modern approaches to error analysis and visualization for nonlinear PDEs, highlighting potential pitfalls and fostering accessible research opportunities.

Findings

Error analysis methods improve understanding of numerical stability.

Visual techniques help verify analytical results.

Open questions encourage undergraduate research in nonlinear PDEs.

Abstract

A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but this can be misleading if the method is not developed and operated carefully due to numerical oscillations, instabilities, and other artifacts of the programmed solution. This chapter provides a friendly introduction to error analysis of numerical methods for nonlinear partial differential equations along with modern approaches to visually demonstrate analytical results with confidence. These techniques are illustrated through several detailed examples with a relevant governing equation, and lead to open questions suggested for undergraduate-accessible research.