Validity of relaxation models arising from numerical schemes for hyperbolic-parabolic systems

TL;DR

This paper validates the convergence of relaxation models derived from numerical schemes for hyperbolic-parabolic systems, demonstrating their effectiveness and proposing a new, more general relaxation model for multi-dimensional systems.

Contribution

The work verifies convergence criteria for existing relaxation models and introduces a new, broadly applicable relaxation model for hyperbolic-parabolic systems.

Findings

Five relaxation models are proven to converge to target systems.

A new relaxation model is proposed and shown to satisfy convergence criteria.

The results support the effectiveness of relaxation-based numerical schemes.

Abstract

This work is concerned with relaxation models arising from numerical schemes for hyperbolic-parabolic systems. Such models are a hyperbolic system with both the hyperbolic part and the stiff source term involving a small positive parameter, and thus are endowed with complicated multiscale properties. Relaxation models are the basis of constructing corresponding numerical schemes and a critical issue is the convergence of their solutions to those of the given target systems, the justification of which is still lacking. In this work, we employ the recently proposed theory for general hyperbolic relaxation systems to validate relaxation models in numerical schemes of hyperbolic-parabolic systems. By verifying the convergence criteria, we demonstrate the convergence, and thereby the approximation validity, of five representative relaxation models, providing a solid basis for the effectiveness of the corresponding numerical schemes. Moreover, we propose a new relaxation model for the general multi-dimensional hyperbolic-parabolic system. With some mild assumptions on the system, we show that the proposed model satisfies the convergence criteria. We remark that the existing relaxation models are constructed only for a special case of hyperbolic-parabolic system, while our new relaxation model is valid for general systems.