Qualitatively accurate spectral schemes for advection and transport

TL;DR

This paper introduces a new spectral discretization method for advection and transport equations that conserves key invariants like positivity and scalar multiplication while maintaining spectral convergence, demonstrated through numerical experiments.

Contribution

A novel spectral discretization technique that conserves invariants of transport and continuum equations, improving upon standard spectral methods.

Findings

Conserves invariants such as positivity and scalar multiplication.

Maintains spectral convergence rates.

Effective in numerical experiments.

Abstract

The transport and continuum equations exhibit a number of conservation laws. For example, scalar multiplication is conserved by the transport equation, while positivity of probabilities is conserved by the continuum equation. Certain discretization techniques, such as particle based methods, conserve these properties, but converge slower than spectral discretization methods on smooth data. Standard spectral discretization methods, on the other hand, do not conserve the invariants of the transport equation and the continuum equation. This article constructs a novel spectral discretization technique that conserves these important invariants while simultaneously preserving spectral convergence rates. The performance of this proposed method is illustrated on several numerical experiments.