Explaining oscillatory behavior in convection-diffusion discretization

TL;DR

This paper investigates the causes of oscillations in convection-diffusion discretizations, introduces methods to eliminate them, and proposes a new error analysis approach to improve the robustness of numerical solutions.

Contribution

It presents a novel error analysis method requiring optimal discrete infinity error and offers strategies to eliminate non-physical oscillations in convection-diffusion problems.

Findings

Oscillations are linked to discretization issues in convection-diffusion problems.

A new error analysis approach improves understanding of solution accuracy.

Discretizing 2D problems benefits from 1D stream line discretization.

Abstract

For a model convection-diffusion problem, we address the presence of oscillatory discrete solutions, and study difficulties in recovering standard approximation results for its solution. We justify the presence of non-physical oscillations and propose ways to eliminate oscillations. A new approach for error analysis that requires establishing optimal discrete infinity error as a first step is introduced and justified. We emphasize that the discretization of two dimensional convection dominated problems benefit from the efficient discretization of the corresponding one dimensional problem along each stream line. Our results are useful in building new and robust discretizations for multi-dimensional convection dominated problems.