An asymptotic-preserving active flux scheme for the hyperbolic heat equation in the diffusive scaling

TL;DR

This paper demonstrates that the active flux scheme with Jacobian splitting is asymptotic-preserving for the hyperbolic heat equation in the diffusive limit, ensuring accurate transition from hyperbolic to parabolic regimes.

Contribution

It proves that the Jacobian splitting-based active flux method is inherently asymptotic-preserving without modifications for the hyperbolic heat equation.

Findings

The AF method is AP in the diffusive scaling.

Numerical experiments confirm the AP property.

The scheme accurately captures the diffusive limit.

Abstract

The Active Flux (AF) method is a compact, high-order finite volume scheme that enhances flexibility by introducing point values at cell interfaces as additional degrees of freedom alongside cell averages. The method of lines is employed here for temporal discretization. A common approach for updating point values relies on the Jacobian Splitting (JS) method, which incorporates upwinding. A key advantage of the AF method over standard finite volume schemes is its structure-preserving property, motivating the investigation of its asymptotic-preserving (AP) behavior in the diffusive scaling. We show that the JS-based AF method without any modification is AP for solving the hyperbolic heat equation, in the sense that the limit scheme is a discretization of the limit heat equation. We use formal asymptotic analysis, discrete Fourier analysis, and numerical experiments to illustrate our findings.