Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

TL;DR

This paper introduces spatially adaptive projective integration schemes for stiff hyperbolic balance laws with spectral gaps, enabling larger time steps and improved efficiency by adapting to spatial variations in relaxation time.

Contribution

The paper develops the first spatially adaptive projective integration schemes tailored for hyperbolic balance laws with spectral gaps, overcoming CFL constraints.

Findings

Schemes achieve large speedup over standard methods

Accurate stability bounds derived for parameters

Numerical tests confirm good accuracy and efficiency

Abstract

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.