Infinite Time Solutions of Numerical Schemes for Advection Problems

TL;DR

This paper introduces a new convergence concept for numerical schemes solving advection problems over infinite time, proving linear methods cannot meet it, but nonlinear jet schemes can, enabling long-term accuracy.

Contribution

The paper defines a novel infinite-time convergence criterion and develops nonlinear jet schemes that satisfy it, advancing long-term numerical solution accuracy.

Findings

Linear methods do not satisfy the new convergence criterion.

Nonlinear jet schemes meet the infinite-time convergence criterion.

Long-time accuracy is improved using the proposed methods.

Abstract

This paper addresses the question whether there are numerical schemes for constant-coefficient advection problems that can yield convergent solutions for an infinite time horizon. The motivation is that such methods may serve as building blocks for long-time accurate solutions in more complex advection-dominated problems. After establishing a new notion of convergence in an infinite time limit of numerical methods, we first show that linear methods cannot meet this convergence criterion. Then we present a new numerical methodology, based on a nonlinear jet scheme framework. We show that these methods do satisfy the new convergence criterion, thus establishing that numerical methods exist that converge on an infinite time horizon, and demonstrate the long-time accuracy gains incurred by this property.